Linköping Studies in Science and Technology Dissertations, No. 1287 Multidimensional MRI of Myocardial Dynamics Acquisition, Reconstruction and Visualization Andreas Sigfridsson Department of Biomedical Engineering Department of Medical and Health Sciences Linköping University, SE-581 85 Linköping, Sweden Linköping, November 2009 Cover: Cardiac deformation in a patient with an acute posterior myocardial infarct, measured using DENSE and visualized using arrow plots of the displacement field. Multidimensional MRI of Myocardial Dynamics Acquisition, Reconstruction and Visualization c 2009 Andreas Sigfridsson, unless otherwise noted Department of Biomedical Engineering Department of Medical and Health Sciences Linköping University SE-581 85 Linköping Sweden ISBN 978-91-7393-494-7 ISSN 0345-7524 Printed 2009 by LiU-Tryck, Linköping, Sweden Abstract Methods for measuring deformation and motion of the human heart in-vivo are crucial in the assessment of cardiac function. Applications ranging from basic physiological research, through early detection of disease to followup studies, all rely on the quality of the measurements of heart dynamics. This thesis presents new improved magnetic resonance imaging methods for acquisition, image reconstruction and visualization of cardiac motion and deformation. As the heart moves and changes shape during the acquisition, synchronization to the heart dynamics is necessary. Here, a method to resolve not only the cardiac cycle but also the respiratory cycle is presented. Combined with volumetric imaging, this produces a five-dimensional data set with two cyclic temporal dimensions. This type of data reveals unique physiological information, such as interventricular coupling in the heart in different phases of the respiratory cycle. The acquisition can also be sensitized to motion, measuring not only the magnitude of the magnetization but also a signal proportional to local velocity or displacement. This allows for quantification of the motion which is especially suitable for functional study of the cardiac deformation. In this work, an evaluation of the influence of several factors on the signal-to-noise ratio is presented for in-vivo displacement encoded imaging. Additionally, an extension of the method to acquire multiple displacement encoded slices in a single breath hold is also presented. Magnetic resonance imaging is usually associated with long scan times, and many methods exist to shorten the acquisition time while maintaining acceptable image quality. One class of such methods involves acquiring only a sparse subset of k -space. A special reconstruction is then necessary in order to obtain an artifact-free image. One family of these reconstruction techniques tailored for dynamic imaging is the k-t BLAST approach, which incorporates data-driven prior knowledge to suppress aliasing artifacts that otherwise occur with the sparse sampling. In this work, an extension of the original k-t BLAST method to two temporal dimensions is presented and iv applied to data acquired with full coverage of the cardio-respiratory cycles. Using this technique, termed k-t2 BLAST, simultaneous reduction of scan time and improved spatial resolution is demonstrated. Further, the loss of temporal fidelity when using the k-t BLAST approach is investigated, and an improved reconstruction is proposed for the application of cardiac function analysis. Visualization is a crucial part of the imaging chain. Scalar data, such as regular anatomical images, are straightforward to display. Myocardial strain and strain-rate, however, are tensor quantities which do not lend themselves to direct visualization. The problem of visualizing the tensor field is approached in this work by combining a local visualization that displays all degrees of freedom for a single tensor with an overview visualization using a scalar field representation of the complete tensor field. The scalar field is obtained by iterated adaptive filtering of a noise field, creating a continuous geometrical representation of the myocardial strain-rate tensor field. The results of the work presented in this thesis provide opportunities for improved imaging of myocardial function, in all areas of the imaging chain; acquisition, reconstruction and visualization. Populärvetenskaplig sammanfattning I kampen mot hjärtsjukdom är det viktigt att kunna mäta hjärtväggens funktion, både för att lära sig hur hjärtat fungerar, för att tidigt kunna bekräfta misstanke om hjärtsjukdom och för att följa upp patienter under behandling eller efter ingrepp. Denna avhandling presenterar ett antal nya metoder för att studera hjärtats rörelse med hjälp av magnetkamera. Med magnetkamera mäter man normalt enbart anatomiska bilder. Det är dock möjligt att modifiera metoden för att även mäta funktionella mått såsom hastighet eller förflyttning. Detta är användbart för att studera hjärtmuskelns funktion. Brus i mätningen är dock ett hinder för att uppnå korrekta mätvärden. I denna avhandling utvärderas hur mycket brus man får när man mäter hjärtväggens förflyttning vid olika mätinställningar. I avhandlingen presenteras även ett sätt att samla in bilder som inte bara beskriver hjärtcykelns variationer, utan även hur andningen påverkar rörelserna. Bilderna man erhåller kan sedan visualiseras genom att låsa läget i hjärtcykeln och istället bara variera andningsläget. Dessa bilder kan då på ett unikt sätt illustrera samspelet mellan höger och vänster kammare, vilket varierar med andningen. Vid bildtagning med magnetkamera är ofta insamlingstiden begränsad; antingen av hur länge en patient kan hålla andan, eller hur länge en patient kan ligga helt stilla. Ett sätt att öka effektiviteten av insamlingen undersöks i denna avhandling. Denna teknik utökas till fallet där både hjärtcykeln och andningscykeln studeras. Vidare presenteras ett sätt att bättre bevara små detaljer i hjärtväggens rörlighet när tekniken används. Ett annat sätt att korta insamlingstiden genom att mäta flera snitt samtidigt presenteras även i avhandlingen. Detta minskar kravet att hålla andan för patienten, vilket gör det möjligt att även mäta på patienter i sämre tillstånd. Slutligen presenteras ett sätt att visualisera den komplexa töjningen av vi hjärtmuskeln. Problemet att visa den samtidiga kompressionen och expansionen i en punkt tacklas genom att visualisera ett strukturerat brusfält. Resultaten presenterade i denna avhandling kan leda till bättre undersökningar av hjärtfunktion, vilket i sin tur möjliggör bättre och tidigare diagnoser, bättre val av behandling och ett mer effektivt sätt att använda de begränsade resurserna som finns inom vården. Acknowledgments I would like to thank my supervisors; Hans Knutsson, Lars Wigström and John-Peder Escobar Kvitting. Hans provided me with a constant stream of ideas and always helped me see things from new perspectives. Lars introduced me to this field and taught me everything about magnetic resonance imaging. John-Peder gave me the vital medical perspective and explained it all in a way even I could understand. Thank you for believing in me and giving me the encouragement I needed. Hajime Sakuma gave me the opportunity to study at his lab in Japan. This changed my life, both professionally and personally, and I am forever grateful. Gunnar Farnebäck and Mats Andersson helped me with much of the A L TEX magic I needed for this thesis, and Henrik Haraldsson gave valuable comments regarding the contents. Ann F. Bolger has encouraged me throughout this whole process. Thank you! I would also like to thank my colleagues at the divisions of clinical physiology and medical informatics, the MRI unit, CMIV and the lab in Japan. You provided me with the atmosphere that I love and the opportunities for interesting discussions on a wide flora of topics. Finally, I wish to thank all friends who put the light in my life when I’m not working. Special thanks to my fellow choir singers and my friends in Japan. Linköping, November 2009 This work has been conducted in collaboration with the Center for Medical Image Science and Visualization (CMIV, http://www.cmiv.liu.se/) at Linköping University, Sweden. CMIV is acknowledged for provision of financial support and access to leading edge research infrastructure. viii Contents Abstract iii Populärvetenskaplig sammanfattning v Acknowledgments vii 1 Introduction 1.1 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . 1.2 Glossary of terms and abbreviations . . . . . . . . . . . . . 2 Cardiac motion 2.1 The cardiac cycle . . . . . . . . . . . . . . . . . . . . . 2.2 The respiratory cycle . . . . . . . . . . . . . . . . . . . 2.2.1 Interventricular coupling . . . . . . . . . . . . . 2.3 Myocardial deformation and tensors . . . . . . . . . . 2.3.1 The stress tensor . . . . . . . . . . . . . . . . . 2.3.2 Eigen decomposition . . . . . . . . . . . . . . . 2.3.3 Strain tensors in Cartesian coordinates . . . . . 2.3.4 The strain-rate tensor . . . . . . . . . . . . . . 2.3.5 The strain tensor in non-Cartesian coordinates 3 Cardiac Magnetic Resonance Imaging 3.1 MRI Principles . . . . . . . . . . . . . 3.2 k -space . . . . . . . . . . . . . . . . . 3.3 k-t sampling . . . . . . . . . . . . . . . 3.4 Temporal resolution . . . . . . . . . . 3.4.1 Prospective cardiac gating . . . 3.4.2 Retrospective cardiac gating . . 3.4.3 TRIADS . . . . . . . . . . . . 3.4.4 Simultaneous resolution of both tory cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . cardiac and . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . respira. . . . . 1 2 3 5 5 6 7 8 9 10 11 12 12 15 15 15 17 19 19 20 20 21 x Contents 3.5 k -space acquisition order . . . . . . . . . . . . . . . . . . . . 4 Motion sensitive MRI 4.1 Velocity measurement . . . . . . 4.2 Displacement measurement . . . 4.2.1 T1 relaxation . . . . . . . 4.2.2 Stimulated anti-echo . . . 4.2.3 Stimulated echo and SNR 4.2.4 Phase reference . . . . . . 4.2.5 DENSE in practice . . . . 4.3 Phase wrapping . . . . . . . . . . 23 . . . . . . . . . . . . . . . . . . . . . . . . 25 25 26 28 29 30 31 31 33 . . . . . . . . . . . . . . in x-f space . . . . . . . . . . . . . . . . . . . . . . . . 35 38 38 39 42 43 6 Tensor field visualization 6.1 Glyph visualization . . . . . . . . . . . . . . . . . . . . . . . 6.2 Noise field filtering . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Adaptive filtering . . . . . . . . . . . . . . . . . . . . 45 45 46 47 7 Summary of papers I: Tensor Field Visualisation using Adaptive Filtering of Noise Fields combined with Glyph Rendering . . . . . . . . . . . . II: Five-dimensional MRI Incorporating Simultaneous Resolution of Cardiac and Respiratory Phases for Volumetric Imaging . III: k-t2 BLAST: Exploiting Spatiotemporal Structure in Simultaneously Cardiac and Respiratory Time-resolved Volumetric Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IV: Improving Temporal Fidelity in k-t BLAST MRI Reconstruction V: Single Breath Hold Multiple Slice DENSE MRI . . . . . . . . VI: In-vivo SNR in DENSE MRI; temporal and regional effects of field strength, receiver coil sensitivity, and flip angle strategies 51 8 Discussion 8.1 Multidimensional imaging . . . . . . . . . . . . . . . . . . . 8.2 Costs of sparse sampling . . . . . . . . . . . . . . . . . . . . 8.2.1 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 56 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Rapid acquisition 5.1 k-t BLAST . . . . . . . . . . . . . . . . . . 5.1.1 The x-f space . . . . . . . . . . . . . 5.1.2 Fast estimation of signal distribution 5.1.3 The k-t BLAST reconstruction filter 5.1.4 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 52 52 53 53 54 Contents 8.3 8.4 8.2.2 Temporal fidelity . . . . . . . . . . . . . . . . . . . . Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Strain and strain-rate estimation . . . . . . . . . . . 8.3.2 Adapting the DENSE acquisition to the application 8.3.3 Optimizing reduction factor versus temporal fidelity 8.3.4 Using k-t2 BLAST for respiratory gating . . . . . . . 8.3.5 Acquisition of velocity data using k-t2 BLAST . . . Potential impact . . . . . . . . . . . . . . . . . . . . . . . . Bibliography xi 57 58 58 59 59 59 59 60 63 xii Contents Chapter 1 Introduction Cardiovascular disease is the leading cause of death in the western world. In Sweden, diseases of the circulatory system were the cause of death of 41% of the men and 42% of the women who died in 2007 . Of these deaths, ischemic heart disease was the cause of death in the largest group, accounting for 20% of male deaths and 16% of female deaths. Decades of research has been conducted to understand the function of the healthy heart, the changes associated with disease, and the causes thereof. Functional studies of the heart wall are the cornerstone for the assessment and follow-up of a large range of cardiac diseases. Cardiac motion and deformation are fundamental properties that are of interest to comprehend the impact of diseases on cardiac function. Quantification of motion and deformation would lead to less subjective assessment and improve our ability to compare the effects of different treatment strategies. The aim of this thesis was to develop methods for measuring deformation and motion of the heart in-vivo with the use of magnetic resonance imaging (MRI). There are challenges in three different areas: • Acquisition Imaging the beating heart requires considerable effort. The time required to obtain an image is often longer than one cardiac cycle. Therefore, signal data from multiple cardiac cycles are typically combined and reconstructed to represent an average cardiac cycle. This requires sophisticated methods for synchronization, especially when respiratory motion is considered. Acquisition efficiency needs to be sufficiently high in order to provide adequate resolution and volumetric coverage within a reasonable scan 2 Introduction time, which is sometimes restricted to that of a single breath hold. This translates into requirements on the pulse sequence itself, as well as rapid sampling of k -space. Measuring the intra-myocardial motion involves sensitizing the phase of the MRI signal to the motion. Using these techniques, myocardial velocity and displacement during the cardiac cycle can be measured. • Reconstruction While image reconstruction in MRI can be as straightforward as a Fourier transform in the simplest case, more advanced approaches are often necessary in cardiac MRI. These range from managing cardiac and respiratory synchronization to alias suppressing reconstruction of sparsely sampled data. Additionally, motion sensitized data often require phase correction to be able to quantify the motion. Finally, computing deformation measures, such as strain or strain-rate tensors, requires additional processing. • Visualization Temporally resolved volumetric data requires the use of advanced visualization techniques. Furthermore, the complexity of myocardial deformation, such as strain and strain-rate, requires methods that can reduce the complexity for easier interpretation. These challenges are addressed in this thesis. 1.1 Outline of the thesis This thesis is organized as follows. In Chapter 2, a brief overview of the motion of the heart during the cardiac and respiratory cycles is provided. Furthermore, the strain and strain-rate tensors that describe the local heart wall deformation are introduced. Principles of cardiac MRI and temporally resolving sampling procedures in particular are presented in Chapter 3. Chapter 4 describes how MRI can be used to quantify motion and deformation. In Chapter 5, ways to shorten acquisition time are discussed, mainly focused on the k-t BLAST method. Visualization of the heart wall deformation strain-rate tensor field is described in Chapter 6. Chapter 7 contains short summaries of the papers that are part of this thesis, and Chapter 8 contains discussion. The papers are included at the end of the thesis. 1.2 Glossary of terms and abbreviations 1.2 Glossary of terms and abbreviations Aliasing DENSE ECG FOV In-vivo k -space k-t BLAST LIC M-mode MRI Myocardium SSFP TRIADS Voxel The replication of a signal caused by periodic sampling. The aliased signal appears at different positions in the corresponding transform space, e.g. at different spatial positions, spatial frequencies, temporal positions or temporal frequencies. Displacement ENcoding with Stimulated Echoes. Electrocardiogram Field of view Within a living organism, as opposed to in-vitro (in the laboratory). The spatial frequency domain of the object being imaged in magnetic resonance imaging. k-t Broad-use Linear Acquisition Speed-up Technique. A method to reduce acquisition time by sparse sampling in k-t space, with subsequent alias suppression during reconstruction. Line Integral Convolution, a vector field visualization technique. Motion mode. Display of dynamics by presenting the temporal dimension on a spatial axis, widely used in ultrasound. Magnetic Resonance Imaging Heart muscle Steady-State Free Precession. An MRI pulse sequence widely used for cardiac imaging. Time-Resolved Imaging with Automatic Data Segmentation. A method for resolving motion with automatic division of the cycle into multiple time frames. Volume element 3 4 Introduction Chapter 2 Cardiac motion There are several types of cardiac motion that are meaningful to study. Quantifying deformation in the cardiac muscle can give useful information about the local function of the muscle. Wall thickening and motion can also give important regional information. The interaction between the left and right ventricles is a more global effect that is influenced by pressure differences external to the heart, such as caused by respiration. 2.1 The cardiac cycle The heart is the organ responsible for pumping blood throughout the body. It is divided into four chambers; the left and right atria and the left and right ventricles, as shown in Figure 2.1. The pump function of the heart is periodic, and the cardiac cycle is divided into two main phases, diastole and systole. In the diastolic phase, the left and right ventricles are filled with blood from the atria through the mitral and tricuspid valves. In the systolic phase, the ventricles contract and blood is ejected through the aortic and pulmonary valves to the aorta and the pulmonary artery. Both the left and right sides of the heart beat approximately simultaneously. The two sides are connected in series with the systemic circuit through the body and the pulmonary circuit through the lungs. Deoxygenated blood from the body is delivered to the right atrium. The blood fills the right ventricle and is subsequently ejected through the pulmonary artery and into the lungs, where it is oxygenated. Oxygenated blood is delivered to the left atrium which then fills the left ventricle. The left ventricle ejects the oxygenated blood through the aortic valve to the aorta, which transports the oxygenated blood to the rest of the body. Common heart rates in healthy persons are 45–80 beats per minute. The 6 Cardiac motion a b RV RA LV LA Figure 2.1: Cardiac configuration in systole (a) and diastole (b) in a fourchamber view. The arrows indicate the right ventricle (RV), right atrium (RA), left ventricle (LV) and left atrium (LA). L denotes left and P denotes posterior directions, respectively. duration of the systolic phase usually varies very little with changing heart rates, but the duration of the diastolic phase can vary substantially . The thickness of the myocardium, the heart wall, of the left ventricle ranges from around 10 mm at the end of diastole to 15 mm at the end of systole . The right ventricular walls are significantly thinner, measuring approximately 5-6 mm. The outer contours of the heart are surprisingly static during the cardiac cycle, as seen in Figure 2.1. Substantial contribution to the volume changes is made by shifting the atrio-ventricular plane in the apex to base direction with the mitral and tricuspid valves open during diastole and in the opposite direction with the valves closed during systole . Wall thickness variations during systole and diastole are commonly measured to assess cardiac motion. Asynchrony of the different parts of the ventricle can also be of interest, especially when studying effects of myocardial ischemia and infarction . 2.2 The respiratory cycle Cardiac motion is highly affected by respiration. The most dominant effect of respiration is the effect on heart position. During inspiration the diaphragm, a muscular interface between the abdominal and thoracic cavities, pulls downward and allows the lungs to expand. The heart is attached to the diaphragm, and is being pulled down during inspiration, as illus- 2.2 The respiratory cycle a 7 b Figure 2.2: Cardiac positions in end expiration (a) and end inspiration (b). The heart is shifted as the diaphragm is pulled down during inspiration. trated in Figure 2.2. Chest muscles also expand the chest cage, but to a lesser extent than the expansion caused by the diaphragm. Typical respiratory rates vary between 10–18 cycles per minute . There is considerable variation between subjects of the respiration, with reports of both a hysteretic relationship between the diaphragm and heart positions as well as shifts in heart position over the respiratory cycle of 25 mm . 2.2.1 Interventricular coupling Respiration also affects the pressure in the thorax. During inspiration, pressure is lowered, to force air from the outside into the lungs. This lowering of pressure reduces resistance in the venae cavae, the veins that transport deoxygenated blood from the body into the right atrium. This in turn increases filling of the right ventricle. At the same time, resistance in the pulmonary system is increased, reducing the filling of the left ventricle. During expiration, the pressure and the corresponding effects are reversed. The interventricular septum acts as a regulator, shifting from one side to the other in order to allow for volume or pressure changes. This shift has been demonstrated by the method presented in Paper II and is shown in Figure 2.3. This is referred to as coupling between the ventricles, and is important in several diseases. Some diseases affect the pericardium surrounding the heart. A stiffer pericardium will exaggerate the interventricular coupling. The ventricular coupling has been demonstrated even without any pericardium, but the effect is reduced . Acute changes in left ventricular function due to abrupt pressure overload of the right ventricle (e.g., from pulmonary embolism) may be explained by interventricular coupling. Long-term right ventricular volume overload, for example caused by pulmonary valve insufficiency, can also be linked to the interventricular interdependence . After open heart surgery, abnormal septal wall motion Cardiac motion 127 LS [mm] Expiration Inspiration 98 97 RV diameter [mm] 128 96 LFW [mm] 32.5 129 32 31.5 31 30.5 30 Expiration Inspiration Expiration Inspiration Expiration Inspiration 88 55 86 84 Expiration Inspiration 36 35 34 33 Expiration Inspiration LV diameter [mm] RS [mm] RFW [mm] 8 54 53 52 51 Figure 2.3: Septal motion over the cardiac cycle, as presented in Paper II. The wall positions of the inner right ventricular free wall (RFW), right side of the septal wall (RS), left side of the septal wall (LS) and inner left ventricular free wall (LWF) were traced through the respiratory cycle in an end diastolic cardiac phase and shown on the left. Note that the septal wall moves towards the right ventricle during expiration and towards the left ventricle during inspiration. On the right, the computed right ventricular (RV) diameter and left ventricular (LV) diameter show RV diameter decreasing during expiration and increasing during inspiration. LV diameter demonstrates the opposite behavior. is commonly observed . The pathophysiological mechanism behind this phenomenon is still disputed [10, 11]. 2.3 Myocardial deformation and tensors Local deformation of the heart wall is an important measure of its function. Damaged myocardium is expected to exhibit less deformation, but it may still show large translation, due to pulling or pushing by neighboring healthy tissue , sometimes referred to as tethering. Therefore, it is beneficial to separate rigid-body motion from shape-changing deformation . 2.3 Myocardial deformation and tensors 9 Deformation and stress of a small volume is usually quantified using a tensor. The general definition of a tensor involves a vector space V and its dual V ∗ , where the elements in V ∗ are linear mappings V → R. A tensor is a multilinear mapping from a number of vector spaces (V ) and/or dual vector spaces (V ∗ ) onto the real space (R). The rank of a tensor is the number of arguments to the mapping. Scalars and vectors are special cases of tensors, namely tensors with ranks 0 and 1, respectively. Consider covariant tensors of rank 2, i.e. bilinear mappings V × V → R. There is a one-to-one correspondence of these tensors with linear mappings V → V ∗ . For vectors v1 , v2 ∈ V and a bilinear mapping g : V × V → R, we can define a new linear mapping f : V → V ∗ as f (v1 ) = g(v1 , ·), where g(v1 , ·) is an element in V ∗ , i.e a linear mapping V → R by v2 7→ g(v1 , v2 ). A metric, or scalar product, defines lengths of vectors. It is a bilinear mapping V × V → R, or equivalently, a linear mapping V → V ∗ . If the vector space V is equipped with a metric, we can thus impose a one-to-one correspondence between elements in V and V ∗ . 2.3.1 The stress tensor To illustrate the concept of a tensor, we may study the stress tensor1 which describes the internal forces acting on a body. One may think in terms of virtually cutting the body along a cut plane. There is a force acting upon this cut plane, not necessarily perpendicular to the plane, but generally having components of shear orthogonal to the plane normal. This is illustrated in Figure 2.4. The stresses, forces per unit area, acting on three orthogonal cut planes are shown on the surfaces of a box. The stresses can be of arbitrary direction, as illustrated by the decomposition into three orthogonal components, one in the normal direction, representing the normal stress, and two in the surface plane, representing shear stress. As the stress tensor is linear, the stresses need only be obtained for three linearly independent cut planes in order to determine the tensor completely. The stress tensor is the linear mapping from the cut plane, which may be represented by its normal (V ∗ ), to the stress (V ∗ ) acting on this cut plane. Or, described in terms of the mapping V ∗ × V → R, how much stress on a cut plane (first argument) there is along a probe vector (second argument). 1 In this thesis, the stress tensor refers to the Cauchy stress tensor, but other stress tensors exist . 10 Cardiac motion Figure 2.4: Illustration of the stress tensor acting on a body. The tensor components are the stresses acting on different cut planes. 2.3.2 Eigen decomposition A useful aid to interpret a tensor is eigen decomposition, after which three (in three dimensions) eigenvectors and corresponding eigenvalues are obtained. The eigen decomposition of the stress tensor is easiest interpreted when viewing the tensor as the mapping V ∗ → V ∗ and representing the cut plane with its normal. An eigenvector e to a mapping T is a vector which is mapped to itself scaled by the eigenvalue λ, i.e. Te = λe with e ∈ V or V ∗ , λ ∈ R. Note that the eigenvalues are only guaranteed to be real for symmetric tensors such as the stress tensor, and in that case the eigenvectors are orthogonal. For the stress tensor, the eigenvectors are the normals to cut planes that contain only normal stress and no shear stress. An example of eigen decomposition is illustrated in Figure 2.5 Figure 2.5: Eigen decomposition of the deformation of a square. The dotted square is the original state and the stippled parallelogram is the deformed state. The arrow on the square indicates shear force. The eigen decomposition is illustrated to the right, with a large lengthening in one diagonal direction and a somewhat smaller shortening in the other diagonal direction. 2.3 Myocardial deformation and tensors 2.3.3 11 Strain tensors in Cartesian coordinates With the stress tensor representing stress, force per unit area, there is a corresponding strain tensor, representing local deformation. The relation between the stress tensor and the strain tensor is modeled using a constitutive law. Strain reflects the shape change between two states, one state usually being a reference state. This requires tracking of myocardial tissue through time, for example by using tagging MRI , DENSE [14, 15], or by point tracking in velocity fields [16, 17]. In Cartesian coordinates, the strain tensors are defined in terms of the deformation gradient F, with the coordinates Fij = ∂xi ∂Xj (2.1) where xi are the coordinates in the deformed state, and Xi are the coordinates in the undeformed state . F itself is not a suitable measure of deformation, because it is sensitive to rigid-body rotation. However, we can define the Lagrangian strain tensor E as2 1 E = (FT F − I) 2 (2.2) with I denoting the identity tensor. The Lagrangian strain tensor is insensitive to rigid-body motion. The Lagrangian strain tensor is convenient when the coordinates xi are known as a function of Xi , as is the case when knowing the material coordinates of over time. If, on the other hand, Xi are given as a function of xi , i.e. the undeformed state is given in the coordinates of the deformed state, another approach is more natural. The inverse deformation gradient, F−1 , given by ∂Xi (2.3) Fij−1 = ∂xj leads to the Eulerian strain tensor e: e= 1 I − (F−1 )T F−1 2 (2.4) Note that the components of the Lagrange and Euler strain tensors do not vary linearly with the extension of the material between the two states. 2 Some texts denote the finite Lagrange and Euler strain tensors by γ and η, and use E and e for the infinite strain approximations where the strain is assumed to be small. Here, we refer to the non-approximate finite strain tensors. 12 2.3.4 Cardiac motion The strain-rate tensor Analogous to how the strain tensors can be derived from the deformation gradient representing the deformation between two states, a strain-rate tensor can be derived from the velocity gradient. The spatial derivative of the measured velocity field, the Jacobian L, has the coordinates Lij = ∂vi ∂xj (2.5) where vi are the coordinates for the velocity vectors. The asymmetric part of the Jacobian contains the rigid body rotation, while the symmetric part is called the strain-rate tensor. The Jacobian is symmetrized according to D= 1 (L + LT ) 2 (2.6) This tensor represents the instantaneous rate of change of strain and has the physical unit s−1 . The strain-rate tensor is commonly used in fluid studies, but may also be applied to studies of myocardial mechanics. The directions of the strain-rate eigenvectors represent the principal directions of lengthening or shortening. The eigenvalues represent the rate of lengthening (positive) or shortening (negative). 2.3.5 The strain tensor in non-Cartesian coordinates By deriving the strain tensor without assuming Cartesian coordinates, different insights can be reached with regard to the meaning of the strain tensor. In a modern description using differential geometry and manifolds the deformation can be described in a particularly elegant way , briefly summarized here for the case of a single point with its tangent space3 . First, denote the undeformed body as an open set B ⊂ R3 and the deformed body as S ⊂ R3 . A smooth, invertible mapping φ : B → S describes the motion between the states, mapping every point in B to its corresponding location in S. Specifically, consider the points X ∈ B and x = φ(X) ∈ S. At each of these points, we have the corresponding tangent vector spaces denoted TX B and Tx S. Each of the tangent spaces are equipped with a metric, mapping tangent vectors to dual tangent vectors, G : TX B → TX∗ B and g : Tx S → Tx∗ S. Recall that metrics define lengths of vectors. 3 For the sake of simplicity, some assumptions are omitted. For a more thorough description, please read the text book referenced. 2.3 Myocardial deformation and tensors g:TxS→Tx*S ϕ* G:TXB→TX*B 13 TxS T XB ϕ* x X ϕ-1 S B ϕ Figure 2.6: Schematic of the manifold representation of the deformation. B is the undeformed body, S is the deformed body and φ denotes the mapping between them. TX B and Tx S are the tangent spaces at the points X ∈ B and x = φ(X) ∈ S, which are equipped with their respective metrics G and g. Through the mapping φ and its inverse, we can define operations pushforward (φ∗ ) and pull-back (φ∗ ) which transform tensors between the tangent spaces. Through φ, we can now define the operations push-forward and pullback, denoted φ∗ and φ∗ , respectively. With these operations, we can transform tensors fields between the manifolds B and S. It turns out that these operations can be defined in terms of the deformation gradient F. The concepts are illustrated in Figure 2.6. The Lagrangian and Eulerian strain tensors, E : TX B → TX B and e : Tx S → Tx S can now be described using their respective associated tensors, which involve the pull-back and push-forward of the metrics for each tangent space: 1 ∗ (φ (g) − G) 2 1 e♭ : Tx S → Tx∗ S = (g − φ∗ (G)) 2 E♭ : TX B → TX∗ B = (2.7) (2.8) E♭ and e♭ are related to E and e through the respective metrics G and g by E♭ = G ◦ E and e♭ = g ◦ e. We see that both strain tensors are described by the difference in metric tensors between the deformed and undeformed states, one of them being pushed forward or pulled back to describe the deformation from the chosen viewpoint. 14 Cardiac motion We further have the relations E♭ = φ∗ (e♭ ) ♭ ♭ e = φ∗ (E ) (2.9) (2.10) which emphasize the symmetry between the two strain tensors, and clarifies how they are defined with their respective points of view in mind. Returning to the Cartesian description, we see the push-forward operation in terms of applying the deformation gradient matrix: E = FT eF 1 E = FT I − (F−1 )T F−1 F 2 1 T E= F F − FT (F−1 )T F−1 F 2 1 T E= F F−I 2 (2.11) Chapter 3 Cardiac Magnetic Resonance Imaging 3.1 MRI Principles MRI is an imaging modality that exploits the nuclear magnetic resonance phenomenon, and is commonly used to produce images of the hydrogen proton distribution in humans. Hydrogen is abundant in the human body in the form of water molecules. MRI incorporates a strong external homogeneous magnetic field, which is used to align the spin distribution of the hydrogen protons along the magnetic field direction. A rotating magnetic field, usually referred to as radio frequency field, is then applied. Tuned to the Larmor frequency of the spins, it is used to tip the spin distribution away from the main magnetic field direction. This tipping is referred to as excitation. After the excitation, the spin distribution undergoes a relaxation process, in which the distribution returns to be directed along the main magnetic field. During this relaxation, the spins emit a signal that is received using induction in coils. During signal reception, additional spatially varying magnetic fields, referred to as the gradients, are applied to encode the spatial position of the signal. The combination of gradient waveforms and rotating magnetic field pulses is called a pulse sequence. 3.2 k -space In MRI, data is naturally acquired in the Fourier domain, which is called k -space [19, 20]. During readout of the MRI signal, which is usually seen as a complex-valued signal, the spatially varying gradients encode a linear phase on the imaging object. A gradient with strength G applied during a 16 Cardiac Magnetic Resonance Imaging time period of t modulates the signal at a location x according to eixγ Rt 0 G(τ )dτ (3.1) where γ is the gyromagnetic ratio of the hydrogen proton. Combined with the fact that the signal is received from the whole object simultaneously γ Rt G(τ )dτ , the signal S follows a familiar and the substitution k = 2π 0 relationship, a Fourier transform: Z S= ρ(x)eix2πk dx (3.2) X where the integral is performed over all spatial positions and ρ is the proton density. This is a highly simplified model, disregarding relaxation, signal decay and spatially varying coil sensitivity during reception, among other things. Nevertheless, it illustrates the Fourier encoding and the role of k -space as the spatial frequency domain. During acquisition, k -space is traversed using different gradient waveforms and the resulting MR signal is sampled. The image is then obtained by a simple Fourier transform of the measured k -space data. As the signal magnitude is decaying during the acquisition, the whole k -space cannot usually be sampled after a single excitation. A common approach is to sample a single line, or profile, from a two or three dimensional k -space after each excitation. For objects of fine structure, high spatial frequencies need to be sampled. This requires sampling of a larger area of k -space using several repetitions and, consequently, a longer acquisition time. Since the spatial frequency domain is being sampled instead of the normal spatial domain, function domain and transform domain can be seen as reversed when compared to conventional signal processing of temporally or spatially sampled signals. Concepts of sampling density and Nyquist aliasing etc. show up in new places. As humans have a finite spatial extent, the spatial Fourier transform is guaranteed to be band limited. This translates into a requirement for the sampling density in k -space to be able to reconstruct the object without aliasing. If k -space is not sampled densely enough, spatial aliasing will occur. This is because regular sampling in the function domain (k -space) will cause periodic repetition of the signal in the reciprocal transform domain (the spatial domain). The sampling can be represented as a multiplication by the Shah-function, III(k), defined as III(k) = ∞ X n=−∞ δ(k − n) (3.3) 3.3 k-t sampling 17 with δ as the Dirac impulse. The convolution theorem states that multiplication of two signals in the function domain corresponds to convolution of their transforms in the transform domain. Since III(k) is self-reciprocal , i.e. it is its own Fourier transform, this means that the transform of the sampled signal is replicated, or aliased, periodically. Furthermore, the similarity theorem states that if a function f (x) has the Fourier transform F (s), 1 then the Fourier transform of f (ax) is |a| F ( as ). This means that the aliased signals get closer to each other with larger sampling distance. If the aliased signals overlap, the true signal can no longer be recovered correctly. 3.3 k-t sampling In dynamic imaging, k -space must be sampled over time as well. Time is discretized in a number of time frames with sufficient rate to capture the dynamics of the object being imaged. The standard method is to sample each k -space position once in every time frame. This can be referred to as regular sampling of the k-t space with full density, shown in Figure 3.1 together with the resulting aliasing of the signal in the x-f space. To reconstruct the data sampled with full density as above, a rectangle function can be used to cut out the transform of the main signal. If data is k x t f Figure 3.1: Regular k-t sampling with full density. Each dot shows a sampling position (left) and the corresponding transforms of the signal (right, white) and aliased signal (right, gray) are separated enough, enabling aliasfree reconstruction. 18 Cardiac Magnetic Resonance Imaging not fully sampled, or equivalently, if the transform is larger than expected, the aliased signals will overlap with the main signal transform, causing reconstruction errors, as shown in Figure 3.2. This is actually quite often the case, especially in the temporal dimension, because the object is seldom bandlimited in the temporal dimension. This is not a big problem, however, because the energy content in the high temporal frequency components is very small compared to in the lower frequency components. k x a t k b f d f x c t Figure 3.2: Regular k-t sampling with half density in the spatial frequency dimension (a) results in overlapping aliased signals (b), causing spatial aliasing errors after reconstruction. Regular k-t sampling with half density in the temporal dimension (c) also results in overlapping aliased signals (d), causing temporal frequency aliasing errors after reconstruction. 3.4 Temporal resolution 3.4 19 Temporal resolution Requirements for spatial and temporal resolution for cardiac imaging often force data acquisition over the course of several heart beats. By assuming that the object undergoes identical motion in each heart beat, different parts of k -space can be sampled in the same cardiac phase but in different cardiac cycles. This approximation is however degraded by respiratory motion and breaks down in case of arrhythmia during the acquisition. There are different methods of controlling k -space acquisition order and keeping track of which parts of k -space have been acquired during the experiment, as described below. 3.4.1 Prospective cardiac gating Prospective cardiac gating , sometimes called triggering, works by alternating monitoring of a cardiac triggering device, such as an electrocardiogram (ECG), and acquisition of k -space data. The acquisition scheme starts by waiting for an R-peak in the ECG, meaning the onset of systole. After the R-peak is detected, the acquisition is delayed for a predetermined time, trigger delay. After the trigger delay, a fixed predetermined number of time frames are collected by acquiring another fixed predetermined number of k -space profiles for each time frame. After acquiring data from all time frames, the acquisition computer returns to monitoring the triggering device. In each successive cardiac cycle, different lines in k -space are acquired. The acquisition is finished when all k -space lines have been acquired. In prospective methods, the time frames are classified already during acquisition, making reconstruction easy. No interpolation is necessary and all cardiac cycles and k -space lines have the same number of time frames. The drawback of this method is its inability to image the later parts of the cardiac cycle, because the number of cardiac time frames acquired needs to be fixed and set small enough to allow the scanner to start monitoring the ECG before the next R-peak. Some variation of cardiac frequency is expected, further limiting the number of cardiac time frames. The advantage is the simplicity of acquisition and reconstruction. This method is often used when only one time frame in a specific phase of the cardiac cycle is acquired, such as in the case of coronary artery magnetic resonance angiography . 20 3.4.2 Cardiac Magnetic Resonance Imaging Retrospective cardiac gating Retrospective cardiac gating , often referred to as cine imaging, solves the problem of imaging the whole cardiac cycle. One common approach incorporates simultaneous acquisition and monitoring of the ECG. The acquisition starts by continuously measuring the first k -space line. When an R-peak is detected, the acquisition advances to the next k -space line. The acquisition is terminated when the whole k -space has been acquired. Instead of only acquiring one k -space line continuously, one can alternate between several, trading temporal resolution for scan time. Also, k -space order is not necessarily linear from top to bottom, but can follow more advanced schemes. Because of variations in heart rate, the number of measurements for each k -space line is not the same for every k -space line. The k -space data is then usually interpolated over time to a number of evenly distributed time frames. This interpolation usually stretches the cardiac cycle linearly, but some more advanced models have been proposed. One such model assumes a constant length systole and stretches diastole linearly, but it has not shown significant improvement over the simple linear model . The benefit of the retrospective method is the ability to resolve the complete cardiac cycle, at the expense of implementation complexity. 3.4.3 TRIADS A method that provides a flexible trade-off between acquisition time and temporal resolution is Time-Resolved Imaging with Automatic Data Segmentation (TRIADS) . Instead of following a fixed scheme for every cardiac cycle, acquisition is adapted to the cardiac phase. TRIADS decides which k -space line to acquire at a given time, in contrast to the cine method, which decides the time(s) to acquire a given k -space line. For every repetition of the TRIADS acquisition, the current cardiac phase is estimated. The estimated cardiac phase is then binned into one of a fixed number of time frames prescribed. TRIADS keeps track of which parts of k -space have already been acquired for each individual time frame, and acquires the next k -space line for the particular time frame. The acquisition continues until a full k -space has been acquired for all time frames. Note that in TRIADS the time frames are not required to come in a predetermined order. In cine imaging, temporal resolution is prescribed by a fixed multiple of the repetition time, which leads to varying number of time frames acquired for each k -space line. In contrast, TRIADS prescribes a number of time frames, and every cardiac cycle is divided into this number of time frames. 3.4 Temporal resolution 21 Acquisition stage Reconstruction stage ky cine ky ky TRIADS ky timeframe timeframe Figure 3.3: An example of cine and TRIADS acquisition schemes. In cine imaging, the acquired profiles (ky ) are changed at each R-peak. In TRIADS, cardiac phase, shown as circles with different shades in this example with four time frames, is estimated for each repetition. Previously acquired profiles are tracked individually for each time frame. Note the variations of RR-intervals. Temporal resolution in absolute time will then vary to be able to fit the number of time frames into the cardiac cycle. A schematic comparison between the cine method and TRIADS is shown in Figure 3.3. Since the binning into time frames is done during acquisition in TRIADS, reconstruction is as simple as for the prospective method. Indeed, one may regard TRIADS as a prospective method, as the binning into time frames usually involves predicting the duration of the current cardiac cycle based on previous cardiac cycles, as opposed to designating time retrospectively. A major difference between TRIADS and prospective gating is TRIADS ability to image the complete cardiac cycle. Also, the cardiac phase estimates can be refined retrospectively, and re-binned using interpolation. This requires that appropriate k -space lines have been acquired at a reasonable number of time points spread over the cardiac cycle. The prospective phase estimates thus still needs to be accurate to some extent. 3.4.4 Simultaneous resolution of both cardiac and respiratory cycles In order to measure cardiac motion affected by respiration, the respiratory cycle needs to be resolved. Since there is still motion during the cardiac cycle, sampling must be synchronized with the cardiac cycle. This can 22 Cardiac Magnetic Resonance Imaging be accomplished by using a prospective triggering approach and acquiring only one time frame per cardiac cycle, but much time is spent waiting for the particular period in every cardiac cycle. By continuously acquiring data, both cardiac and respiratory cycles can be resolved simultaneously. In other words, a full image or volume is acquired for every combination of cardiac phase and respiratory phase. This adds a new dimension to cardiac imaging; being able to freeze motion during the cardiac cycle and visualize the effects induced by respiration on cardiac function. With simultaneous resolution of both cardiac and respiratory cycles, the time line becomes a two-dimensional time plane. If the cardiac and respiratory cycles are fully covered, as when using the TRIADS method, both dimensions are cyclic. The topology of the temporal dimensions can then be visualized as a torus, as shown in Figure 3.4. Even though the individual temporal dimensions are cyclic, their combination in actual time is more complex. This makes the cine and prospective methods unsuitable for acquiring data resolved to both dimensions simultaneously, unless the respiration rate can be controlled . The TRIADS method, however, only requires that the phases in the individual cycles can be estimated. Every repetition in the acquisition then involves estimating both cardiac and respiratory phase, classifying them into a combined time frame, and the TRIADS scheme takes care of filling the k -space in every time frame. Acquisition of simultaneously resolved cardiac and respiratory cycles in a two-dimensional slice has been presented previously . In that work, TRIADS was used to resolve the respiratory cycle, but within each cardiac cycle, retrospective cine imaging was performed. This caused the respiratory phase estimates made at the beginning of every cardiac cycle to be Respiratory time Cardiac time Cardiac time Respiratory time Figure 3.4: Simultaneous resolution of both cardiac and respiratory cycles gives a two-dimensional temporal plane (left). Since the plane is cyclic in both dimensions, the topology can be visualized as a torus (right). 3.5 k -space acquisition order 23 assumed constant throughout that cardiac cycle. In Paper II, a volumetric method is presented, extending TRIADS to two simultaneous temporal dimensions. 3.5 k -space acquisition order In cardiac imaging, balanced steady-state free precession (SSFP)  is a frequently used pulse sequence. It provides strong signal from the blood and allows for short repetition times with maintained signal level. This comes at the requirement of a fast gradient switching system and a stable homogeneous magnetic field. Gradient systems that are fast enough are readily available, but disturbances in the magnetic field can in some cases be a problem. One cause of problem is eddy currents disrupting the steady state. These eddy currents can be caused by large changes in phase encoding gradient strength between successive excitations , i.e. large jumps in k -space. These effects can be removed by acquiring the same k -space line twice in two successive excitations and taking the complex average . This will, however, double the acquisition time. Another way to reduce the effects is to minimize the jumps in k -space by choosing an appropriate acquisition order. For prospective and retrospectively gated acquisitions, this is easy, since the k -space order can be controlled directly, and jumps can be minimized by choosing a zig-zag pattern. In TRIADS, however, the already acquired parts of k -space are generally different for different time frames. Time frames may be acquired in a non-predictable order, especially when resolving two independent temporal dimensions. Furthermore, the time between excitations is very short, imposing a limit to how much computation can be performed in order to optimize the acquisition order in runtime. In Paper II, this is solved by using a predefined k -space profile order curve and keeping a time-frame local progress counter that indicates how many lines along this profile order curve have been acquired for that particular time frame. The profile order curve is a discrete mapping from the onedimensional progress counter to the two-dimensional ky − kz space. The kx dimension is covered by reading a whole line in k -space for each repetition. For the acquisition parameters used in Paper II and III, the time spent in each time frame is on the order of 10–15 excitations until the time frame is changed. Since all timeframes are approximately equally common, the differences between progress counters are expected to be small. This imposes three design criteria on the profile order curve: • Each point in the ky − kz plane should be visited exactly once. 24 Cardiac Magnetic Resonance Imaging • Two subsequent points along the curve should be adjacent to each other in ky − kz space. • The distance in ky − kz space between two fairly close points on the curve should be minimized. A curve which addresses these design goals is the Hilbert curve, proposed by David Hilbert in 1891. The locality of the Hilbert curve is close to optimal, expressed in terms of the maximum value of |Hilbert(p1 ) − Hilbert(p2 )|2 |p1 − p2 | (3.4) which has a low bound . The squared distance in the numerator is computed in ky − kz space and the distance in the denominator is the distance along the curve for two different points p1 and p2 . This means that close points along the curve are also close in the ky − kz space. Thus, when the time frame differs between excitations, the jump in k -space will be kept short. A first order Hilbert curve consists of a single U-shape as seen in Figure 3.5a. Subsequent levels are generated by replacing the U-shape with four rotated versions linked together with three joints (Figure 3.5b-d). a b c d Figure 3.5: A Hilbert plane filling curve can be used to control acquisition order to reduce eddy current effects in balanced SSFP imaging. It is constructed recursively, and levels 1 through 4 are shown in a-d. Chapter 4 Motion sensitive MRI MRI is an extremely flexible tool when it comes to motion sensitivity. It is possible to measure velocities in arbitrary directions [32, 33, 34], displacement , elasticity  as well as anisotropic diffusion . These techniques are useful for blood flow measurements as well as myocardial motion analysis . Combination of diffusion and phase-contrast frameworks has made it possible to measure the intra-voxel velocity standard deviation , a measure of turbulence intensity of the flow. All these methods employ the concept of a complex-valued MRI signal, where the phase of this signal can be sensitized to motion. 4.1 Velocity measurement Measuring velocity using MRI is done by encoding the velocity into the phase of the complex MRI signal through the use of a so-called bipolar gradient pulse, consisting of two consecutive pulses in opposite directions. Considering the lobes separately, the concept can be explained as an encoding of the position of each spin into its phase at the first lobe, and at a moment later in time, subtracting the phase corresponding to its new position. The resulting phase is then proportional to the difference in position during the interval of the pulses. The position encoding in the phase is seen by noticing the change of Larmor frequency when the gradient is applied. Integrated over time t, this change of frequency is translated into a change of phase ϕ: ω = γG(t)x(t) Z t ϕ(t) = ω(τ )dτ 0 (4.1) (4.2) 26 Motion sensitive MRI where ω is the frequency, γ is the gyromagnetic ratio, G(t) is the gradient and x(t) is the spin position. Expressing the position over time in terms of starting position x0 , velocity v, acceleration a etc.: 1 x(t) = x0 + vt + at2 + · · · 2 (4.3) we see that the phase shift caused by the constant part of velocity is proportional to the first moment of the gradient pulse. ϕ(t) = γ Z t G(τ )x(τ )dτ Z t Z t Z t 1 2 = x0 γ G(τ )dτ + v γ G(τ )τ dτ + a γ G(τ )τ dτ + · · · 2 0 0 0 (4.4) 0 The phase is thus proportional to the first moment of the gradient waveform. Incidentally, the case of non-constant velocity, the influence in phase of acceleration is proportional to the second moment of the waveform, and so on. 4.2 Displacement measurement Displacement measurement is in principle very close to velocity measurement. By considering the velocity measurement as a displacement measurement over a short period of time, the extension is straightforward. By inserting a pause between the two lobes of the bipolar gradient, the corresponding displacement is encoded into the phase. In practice, however, a problem arises when trying to separate the lobes of the bipolar gradient waveform. During the time between the pulses, the signal decays with T2∗ relaxation, which in the myocardium is in the order of 10 ms. A special MR technique can be used, where the signal is stored in the longitudinal magnetization. There, limited only by T1 relaxation which is in the order of 1 s, displacement can be measured over much longer time. This approach is called a stimulated echo, and its use with displacement measurement is known by its MR-acronym DENSE (Displacement ENcoding using Stimulated Echoes) . The commonly used approach in DENSE involves encoding the position at the time of the R-wave in the ECG using a so called 1-1 SPAMM (SPAtial Modulation of Magnetization) preparation encoding. At a later 4.2 Displacement measurement 27 ECG 90x 90x α RF RF Gslice Gphase Gro Gro Position encode Crusher 1-1 SPAMM preparation Position decode Readout gradients Image readout Figure 4.1: An illustration of a DENSE pulse sequence. A 1-1 SPAMM preparation module encodes the position into the magnetization of the spins. Later in the cardiac cycle the position is decoded and an image is read out. time point in the cardiac cycle, for example at the end of systole, the position is decoded. The acquired image then contains the displacement that occurred during the entire systolic period. Studying systolic motion using velocity measurement based approaches requires integration or tracking of the myocardium during a time-resolved image sequence which accumulates errors and noise over time. Using DENSE, the systolic deformation can be acquired directly, in a single image. If the deformation progression during this time period is desired, DENSE can also be acquired in a cine fashion, with multiple read-outs of the stimulated echo during the cardiac cycle . One example of a DENSE pulse sequence is illustrated in Figure 4.1. Pulse sequence wise, the DENSE approach is very similar to tagging MRI . In fact, using the harmonic phase (HARP) analysis , the techniques are conceptually the same . The difference is that in DENSE a position decoding gradient is used whereas in tagging MRI stripes are still part of the image and not removed until the HARP post processing step. The use of a stimulated echo in DENSE splits the bipolar gradient pulse between two excitation pulses. The role of the excitation pulses is to store the magnetization in the longitudinal direction to make it less sensitive to relaxation. This does not, however, store the full phase in the longitudinal direction, but only the cosine of the phase. Consider the first 90◦ pulse that is used to tip all magnetization into the transverse plane, before the position encoding. Then, a monopolar gradient is used to encode the position into the phase of the spins. The spins now have a distribution onto the 28 Motion sensitive MRI entire transverse plane. The second 90◦ pulse, assumed here to be around the x-axis, is used to rotate the spins from the XY plane to the XZ plane, i.e. partly transversal and partly longitudinal. Due to T2∗ relaxation and/or after crushing the signal with a strong gradient, the remaining transverse component vanishes. This results in a cosine encoding of the phase, as the magnetization is projected onto the longitudinal axis. The tissue containing the cosine modulated magnetization is now subjected to the motion of the heart. At a desired point in time, a third RF pulse is used to transfer the magnetization onto the transverse plane for readout. Since the magnetization is the cosine of the phase, reconstruction of the original phase can seem troublesome. The trick here is to see the signal in the shape of an Euler decomposition of the cosine function: cos(ω) = 1 −iω e + eiω 2 (4.5) The two complex exponentials will be separated in k -space and are sometimes referred to as the stimulated echo and the stimulated anti-echo. This split into two parts implies a signal loss of 50% since only one of the echoes is imaged. 4.2.1 T1 relaxation Using the stimulated echo technique requires important considerations regarding T1 relaxation. While storing the complex signal in the longitudinal magnetization makes it invulnerable to T2 relaxation, T1 relaxation is inevitable. During T1 relaxation the stimulated echo gradually returns to the equilibrium state which is not position encoded. At the time of image read-out, both the stimulated echo signal and the T1 relaxed signal will be present, resulting in what is sometimes referred to as a T1 artifact appearing as stripes across the image. The T1 artifact can be suppressed using different approaches, or a combination thereof. The CSPAMM (Complementary SPAMM) approach used in tagging is based on a two-experiment acquisition . In one of the acquisitions, the sign of one of the RF pulses during encoding is reversed. This results in a sign reversal of the stimulated echo, but no sign reversal of the T1 relaxed signal. A subtraction between the two images cancels the artifact and doubles the signal. The concept is illustrated in Figure 4.2. If there is motion between the acquisitions with reversed RF signs, such as a slight shift of diaphragm position during a long breath hold, the T1 artifact will not be completely removed. Another approach utilizes the fact that the T1 relaxed signal and the stimulated echo are separated in k -space. By 4.2 Displacement measurement a b 29 c Figure 4.2: Using the CSPAMM technique, the T1 artifact is reduced. Two images (a, b) are acquired using different polarity of the second RF pulse in the 1-1 SPAMM preparation sequence. The stimulated echo will then have opposing signs, whereas the T1 relaxed signal will not. By subtracting them (c), the stimulated echo will double and the T1 signals will cancel. Only the magnitude of the complex images is shown. acquiring only a small window centered on the stimulated echo, the influence of the T1 relaxed signal should be small. This, however, results in an impaired spatial resolution of the stimulated echo signal. Further signal separation can be achieved by applying stronger encoding and decoding gradients, but this might result in signal loss due to strain induced phase dispersion . The separation of peaks in k -space can also be utilized in a k -space based filter, suppressing the main lobe of the T1 signal. Other approaches are also pursued, such as using through-plane signal modulation  or an inversion pulse nulling signal with a specific T1 relaxation time . 4.2.2 Stimulated anti-echo The stimulated anti-echo also causes artifacts in DENSE. As with the T1 relaxation artifact, the signal is shifted in k -space. Similar techniques exist for suppressing this artifact. By using strong displacement encoding, the stimulated anti-echo can be shifted sufficiently far out in k -space to limit its influence, as illustrated in Figure 4.3. Again, strain induced dephasing limits how far the anti-echo can be shifted in practice. Generalizing the CSPAMM-concept allows for subtraction of the anti-echo [47, 48, 49]. Alternatively, if the anti-echo signal is acquired in its entirety, it can be used in the image reconstruction and intrinsically serve as a phase reference . 30 Motion sensitive MRI ky a b T1 relaxed signal ky T1 relaxed signal kx Stimulated anti-echo Stimulated echo kx Stimulated anti-echo Stimulated echo Figure 4.3: Illustration of the removal of the stimulated anti-echo. The k space appearance right before application of the displacement decoding gradient (a) shows three major signal components; the stimulated anti-echo, the T1 relaxed signal and the stimulated echo. The T1 relaxed signal can often be quite strong, illustrated here by the larger star. After the displacement decoding gradient, all echoes are shifted in k -space (b). The sampled part of k -space is indicated by the grey rectangle. In this way, the stimulated anti-echo is removed. The T1 relaxed signal is removed by other means, as described in the text. 4.2.3 Stimulated echo and SNR T1 relaxation not only causes image artifacts, it also consumes valuable stimulated echo signal. Moreover, excitation consumes part of the stimulated echo. Consumed or relaxed stimulated echo is unusable until the next cardiac cycle. For multi-phase tagging and DENSE, a common approach is therefore to vary the flip angle to obtain constant signal level of the stimulated echo [43, 51]. The flip angle is increased with each excitation to compensate for the loss from the previous excitation and T1 relaxation. The flip angles are determined by αk−1 = arctan sin(αk )e(−(tk −tk−1 )/T1 ) (4.6) where αk and tk are the flip angle and time for the k:th excitation, respectively. This is solved backwards, and the final flip angle can be optimized with respect to the heart rate and myocardial T1 and the relaxation time available . This scheme will provide the maximum constant SNR during the cardiac cycle, which has been assumed to be the best choice. However, as is seen in Paper VI, the resulting constant SNR can be surprisingly low in practice. For applications where the systolic deformation is most important, a fixed flip angle was found to provide superior SNR. 4.2 Displacement measurement a b 31 c Figure 4.4: Two images acquired using different displacement encoding directions (a, b). After phase subtraction (c), all phase errors common to both acquisitions are canceled. Here, the complex valued images are displayed by mapping magnitude to intensity and phase to hue. 4.2.4 Phase reference Ideally, the phase in an image acquired using DENSE should be proportional to the displacement. Due to several effects, including B0 inhomogeneity, a phase error will remain. This is usually compensated for with the use of a phase reference scan, whereby several images are acquired using different displacement encodings. All phase errors not arising from the displacement encoding gradients themselves will then cancel after phase subtraction, as illustrated in Figure 4.4. There are many different ways of encoding the displacement in DENSE, which can be adapted to include a phase reference [53, 54]. As mentioned above, the stimulated anti-echo may also be used to correct phase errors . 4.2.5 DENSE in practice Figure 4.5 shows how DENSE can be used in practice. A patient with an acute posterior myocardial infarct was examined using MRI. Delayed gadolinium enhancement showed the extent of non-viable myocardium in the posterior region, and a T2 weighted image showed edema in the same region. Using DENSE, reduced motion and strain is evident in the infarcted region. 32 Motion sensitive MRI Delayed enhancement T2 weighted 0.5 0 -0.5 Euler strain eigenvalue e2 Euler strain eigenvalue e1 Figure 4.5: MRI data from patient with an acute posterior myocardial infarction. The arrow map computed using DENSE shows reduced motion in the posterior region. Delayed enhancement, indicating non-viable myocardium, and T2 enhancement, indicating edema, is also visible in the same region. Eulerian strain eigenvalue maps indicate reduced strain in the infarcted region. 4.3 Phase wrapping 4.3 33 Phase wrapping When displacement or velocity is encoded into the phase of the complex MRI signal, an ambiguity arises due to the cyclic behavior of phase accumulation during the influence of the gradient fields. The phase ϕ is encoded according to ϕ = γM1 v mod 2π (4.7) for a velocity or displacement v, a gradient first moment of M1 and a gyromagnetic ratio of γ. Decoding results in an ambiguity because of the unknown n: ϕ + n2π ,n ∈ Z (4.8) ṽ = γM1 Manual correction of these effects is feasible but tedious for larger data sets, such as multi slice or multi phase. Various methods for automatic or semi automatic correction of these exist [55, 56, 57]. In many phase-contrast applications, the encoding strength is adapted to the expected maximum velocity in order to avoid the phase wrapping problem. In DENSE, however, the encoding strength is often chosen high enough to provide separation of the anti-echo and the T1 relaxed signal, which in practice results in considerable phase wrapping. For strain analysis, as opposed to actual displacement measurements, only a local region of no phase wrapping is needed. This can simplify the operation significantly, since the absolute displacement is not needed. 34 Motion sensitive MRI Chapter 5 Rapid acquisition The demands for spatial and temporal resolution in cardiac MRI are usually not compatible with the desired acquisition time. Spatial resolution may be improved by acquiring more of k -space, at the expense of increased scan time and/or decreased signal-to-noise ratio. Scan time may be reduced by decreasing temporal resolution, which is often not desirable. Increase of temporal resolution is also usually limited by the shortest repetition time available. Much effort has been put into reducing scan time while maintaining spatiotemporal resolution. One category of improvements is pulse sequence design for faster acquisition of the same amount of data. Echo-planar methods acquire a whole plane of k -space in one or a few excitations . These methods are sensitive to field inhomogeneities, chemical shift effects and signal decay during the long read-out. Gradient pulse optimization can be used to some extent to reduce the repetition time, but ultimately, gradient hardware or peripheral nerve stimulation caused by rapid gradient switching sets a limit. Another way of reducing acquisition time is to collect fewer points in k -space. By exploiting spatiotemporal structure of the object being imaged, essentially the same images can be reconstructed from less data. Scan time is reduced by a so-called reduction factor. One should bear in mind, though, that almost all of these acquisition time reduction techniques come at the cost of increased noise or artifacts in the reconstructed image. Modeling of the signal using various kinds of priors, thereby fitting the actual data to the implied model, is commonly used. This model fit is obviously erroneous if the data does not conform to the model. The difficulty lies in finding good models which can also be exploited in MRI. Below is a short list of common methods to shorten acquisition time. 36 Rapid acquisition Partial Fourier imaging Traditional Fourier encoding consists of acquiring a Cartesian sampling of k -space with sufficient sampling density to avoid spatial aliasing. A k -space is acquired that covers spatial frequencies high enough to encode the desired resolution. After a fast Fourier transform, a complex image is reconstructed. The image should ideally be real, which is equivalent to a Hermitian symmetry in k -space. Half of k space could therefore be reconstructed from the other half, eliminating the need for acquiring a symmetric k -space. In practice, the image is not real, but some phase variations are present, mainly due to inhomogeneities in the magnetic field, caused by susceptibility effects. These phase variations usually vary slowly over the image and by acquiring slightly more than half of k -space (typically 55–65%), the phase variations can be reconstructed from the symmetric part of k -space and removed from the data . Non-Cartesian encoding It is not necessary to acquire a Cartesian sampling of k -space. Other sampling schemes may be beneficial. Instead of acquiring a rectangular k -space, a circular one can be acquired, having the same spatial resolution in all directions. This eliminates the need to acquire the corners of k -space. Spiral read-out trajectories  instead of conventional ones can cover larger parts k -space per repetition, and is sometimes referred to as echo-planar imaging methods. Radial and spiral sampling schemes also show more visually forgiving aliasing artifacts when using undersampling than Cartesian sampling. Non-Cartesian sampling requires more complicated reconstruction, however, typically involving a process called gridding , or a non-uniform Fourier transform  which may introduce small errors. HYPR Projection imaging has gained much interest, because of the forgiving appearance when using large undersampling factors and thus rapid image acquisition. HighlY constrained backPRojection (HYPR)  has demonstrated an impressive reduction factor of 225 for time-resolved imaging. Temporal averaging is used to reconstruct a composite image, which is then used to constrain backprojections of individual radial read-outs, depositing the projection data only in the objects being imaged. This requires, however, that the objects in the imaging volume are sparse and do not change position over time. Thus, while it might be useful for contrast enhanced angiography, it is not directly applicable for imaging of cardiac motion. Parallel imaging By exploiting the low-frequency spatial encoding and simultaneous signal reception of multiple surface coils, parallel imaging methods such as SENSitivity Encoding (SENSE)  or GeneRalized Autocalibrating Partially 37 Parallel Acquisition (GRAPPA)  can be used to decrease scan time. In these methods, k -space is undersampled, causing spatial alias overlap. This overlap can be recovered since there are several measurements by the individual coil elements, and the aliased signal components are encoded with different coil sensitivities. Acquisition may be shortened by a reduction factor up to the number of coils used in signal reception, but noise becomes a problem when using high reduction factors. This is caused by signal and noise correlation between the coils; the coils essentially see the same signal when using a large number of coils. Typical reduction factors when using SENSE are 2–4. Keyhole, BRISK, TRICKS Keyhole , Block Regional Interpolation Scheme for k -space (BRISK)  and Time-Resolved Imaging of Contrast Kinetics (TRICKS)  are techniques that use varying temporal sampling density for different parts in k -space. The central lines are typically acquired every time frame and the outer k-space lines are acquired more seldom, e.g. every second or third time frame. The idea is that the main part of the image contrast lies in the center of k-space. The signal model assumes that the dynamic information has low spatial frequency, which is not valid for moving edges but may be useful in contrast enhanced angiography. Reduced field of view Reduced field of view (RFOV) [69, 70] assumes that the field of view can be divided into a static region and a dynamic region. A fully sampled k -space can then be acquired for one time frame, while dynamic imaging can be limited to the smaller dynamic part of the field of view. Spatial aliasing overlap will occur in the dynamic data, but can be recovered because the static data can be estimated. UNFOLD Unaliasing by Fourier-Encoding the Overlaps Using the Temporal Dimension (UNFOLD)  samples the k-t space in an interlaced fashion; odd k -space lines are sampled in odd time frames and even k -space lines in even time frames. If the time frames are reconstructed individually, spatial aliasing overlap will occur, due to the undersampling. The aliasing signal will, however, appear with alternating phase between the time frames and can be filtered out. This is, in principle, an extension of the model used in RFOV imaging. Instead of dividing the FOV in an entirely static and a fully dynamic region, some motion is allowed in the static region. The FOV is thus divided into a high dynamic region and a low dynamic region. If the regions are exactly one half of the FOV apart, the particular undersampling in k-t space can be seen as overlapping the low dynamic region with 38 Rapid acquisition the high dynamic region, but with one of the regions shifted in temporal frequency by one half of the temporal bandwidth. In this sense, the full temporal sampling bandwidth can be shared between the two regions. By extending the sampling bandwidth by as much bandwidth as is contained in the low dynamic region, both signals will fit without aliasing. The extra bandwidth needed is usually much less than the factor of 2 gained by the undersampling. UNFOLD can also be seen as a special case of k-t BLAST, described below. k-t BLAST and k-t SENSE A method for dynamic imaging that has gained much attention over the last few years is k-t BLAST (Broad-use Linear Acquisition Speed-up Technique) . One of the reasons for its popularity is the high achievable reduction factors. Two-dimensional and three-dimensional acquisitions with reduction factors of 5 or 8 have been presented with very high image quality. The principle of the method will be described in detail in Section 5.1. The k-t BLAST approach can also incorporate multiple coils and their spatial sensitivity to improve reconstruction. This is called k-t SENSE, which is a multiple-coil extension of k-t BLAST, rather than simultaneous independent use of k-t BLAST and SENSE. Other variants and hybrids exist, with names such as TSENSE, TGRAPPA, k-t GRAPPA, FOCUSS and UNFOLD-SENSE . The k-t BLAST approach can be improved upon by using a PCAconstrained reconstruction . A set of spatially invariant principal component basis functions is found from the fully sampled training data, and used to constrain the reconstruction of the aliased data. 5.1 5.1.1 k-t BLAST The x-f space As described in Section 3.3, when sampling the k-t space regularly, signal aliasing is introduced in the reciprocal x-f space. In typical cardiac imaging contexts, the signal content in x-f space has a localized appearance, i.eṁany spatial positions have a narrow temporal bandwidth. This is shown in Figure 5.1. Because one can control the sampling of k-t space, one can also control how the aliased signals are packed in the x-f space. It is not necessary to sample the k-t space on an axis aligned grid. A sheared grid, forming a lattice, is still regular and will create periodic aliasing in the x-f space. The locations of the aliased signals will then also form a lattice. Using lattice sampling, the localized aliased signals can be packed tighter, enabling 5.1 k-t BLAST a 39 b c x f Figure 5.1: One time frame from a 2D cardiac acquisition (a), from which one column has been selected (white line). The column’s development over time (b) has a signal distribution in x-f space that is highly localized (c). reduced sampling density in k-t space and thus faster acquisition. Two examples of k-t lattice sampling and the corresponding signal packing are shown in Figure 5.2. UNFOLD uses this principle, because the interlaced sampling pattern used in UNFOLD is also a lattice. A fixed predetermined filter is then used to recover the signal. The filter is a temporal low-pass filter with different bandwidths for the high and low dynamic regions. The use of a fixed filter imposes a limit on the packing, because the bandwidth of the filter in the whole high-dynamic region must be wide enough to capture the dynamics of the highest bandwidth in the region. The k-t BLAST approach uses a filter more closely matched to the actual data, enabling potentially tighter packing and thus higher reduction factors. Furthermore, this filter can be used to suppress signal where the aliased signals dominate over the main signal. This involves obtaining an estimate of the main signal distribution without aliasing. This estimate can be used to further improve upon UNFOLD by using a reconstruction filter adapted to the measured data. 5.1.2 Fast estimation of signal distribution in x-f space Typical x-f signal distributions of cardiac objects have an additional property that can be exploited. The temporal bandwidth varies reasonably slowly over the spatial dimensions. Especially, regions with low temporal bandwidth usually form large continuous regions. An estimate of the signal distribution does not need to be of high spatial resolution , but only indicate where there is strong signal with high temporal bandwidth. The signal distribution estimate, sometimes referred to as training data, can thus be measured by sampling only a few of the central lines in k -space. 40 Rapid acquisition k x a t k b f d f x c t Figure 5.2: Dense sampling on a k-t lattice (a) and the corresponding signal packing in x-f space (b). Tight signal packing with no overlapping ensures alias free reconstruction. Too sparse lattice sampling in k-t space (c), on the other hand, packs signals too tight in x-f space causing overlapping and thus prohibits correct reconstruction. Because an alias-free estimate is desired, these lines have to be sampled with full temporal bandwidth, i.e. sampled in every time frame. A schematic illustration of the k-t BLAST approach is shown in Figure 5.3. The k-t space is sampled using a sparse lattice, containing high resolution data with aliasing, and a dense region in the center of k -space in all time frames, containing information to estimate the signal distribution in x-f space. Since the sampling pattern is known, the positions of the aliased signals are also known. By knowing both an estimate of the true and the aliased signal distributions, one can suppress aliasing in the signal reconstruction. 5.1 k-t BLAST Acquisition matrix 41 Low resolution signal estimate k x t f x x f High resolution data with aliasing x f k−t BLAST Reconstruction Estimated aliased signal x f Filter output t Reconstructed data Figure 5.3: Schematic illustration of the k-t BLAST approach. The k-t space is sampled in two ways. Central lines in k -space (triangles and black circles) are fully sampled, yielding a low resolution estimate of the signal distribution. The sparse lattice (triangles and hollow circles) yields high resolution data with aliasing. An estimate of the aliased signal can be obtained from the signal estimate and sampling lattice. Through Wiener filtering, the aliasing can be suppressed, and a Fourier transform in time yields the final output. The data used in this figure is the same as in Figure 5.1. 42 5.1.3 Rapid acquisition The k-t BLAST reconstruction filter The measured signal from the lattice sample points is a sum of the true signal and aliased signals, originating from other spatial locations and temporal frequencies. By treating the signal in each spatial position as widesense stationary in time, a Wiener filter approach can be used to filter out the aliased signal. Furthermore, measurement noise is also expected, so the Wiener filter becomes M2 + P M2 2 Malias + Ψ2 (5.1) P 2 where M 2 is the signal distribution estimate, Malias is the estimated 2 aliased energy as shown in Figure 5.3 and Ψ is the measurement noise variance. The k-t BLAST reconstruction filter can be seen as a quotient between the desired signal energy estimate and the measured signal energy estimate, consisting of the sum of the true signal, the aliased signal and the measurement noise. Summing the signals in this way is appropriate for uncorrelated signals. The aliased signal is from different spatial locations and temporal frequencies, by lattice design, and correlation is thus expected to be very low. The measurement noise is mainly caused by thermal noise emitted by the subject and in the receiver electronics. The noise is Gaussian white , as long as one considers the complex signal measured. The measurement noise variance can be obtained by measuring the variance for a homogeneous region in the image, such as the background. For reconstruction of a wide-sense stationary source with wide-sense stationary noise, the Wiener filter is the optimal linear estimator in the least squares sense . The filtering process is intuitively performed by multiplication in the x-f space, but a corresponding convolution filter kernel in k-t space can also be considered. Such a filter would fill in the blank positions in the k-t space which are not on the sampling lattice. The effect of the filter varies depending on the amount of signal overlap in x-f space. In areas with no overlap, the filter reduces to an ordinary noise suppressing Wiener filter, passing frequencies where the signal is dominant over the measurement noise. Where there is no true signal, the aliasing and noise are efficiently removed. Where there is overlap, the filter will favor the dominant signal, using a weight corresponding to the relative strengths of true signal and aliasing. When the signals in x-f space overlap, which is difficult to avoid, some reconstruction error is unavoidable. The use of the Wiener filter minimizes this reconstruction error in the least squares sense, but there is still a re- 5.1 k-t BLAST 43 construction error. However, the least squares error norm is not necessarily the best norm for motion analysis. The reconstruction error will tend to suppress high temporal frequency signal when it overlaps with strong signal of low temporal frequency. In Paper IV, a modified version of the filter described in Eq. 5.1 is evaluated. In that setting of studying the cardiac wall motion of a patient with ischemic heart disease, an improved depiction of the wall motion was found, at the expense of slightly increased aliasing and noise. 5.1.4 Implementation details Lattice optimization The locations of the aliased signals are determined by the sampling lattice in k-t space. Thus, different sampling lattices can cause differing amounts of signal overlap and different reconstruction performance. Maximizing the minimum distance between the signal aliases in x-f space is one way to reduce the signal overlap for a given field of view, temporal resolution and reduction factor . Maximizing minimum alias distance, however, does not guarantee minimum overlap. To minimize overlap, the actual signal distribution for the particular acquisition needs to be taken into account, which is a much larger problem. The prescribed lattice is not always realized in practice, for example when using TRIADS or retrospective gating. Errors in the estimation of cardiac or respiratory phase may result in sampling locations in k-t space that do not form a lattice. Reconstruction from arbitrary sampling in k-t space has been derived, but direct analytic solution to the problem was deemed infeasible due to computational complexity . Iterative solution to this problem has been presented , enabling the use of non-Cartesian k-t BLAST. Baseline subtraction The temporal variation of the signal can be modeled as a deviation from a baseline signal. This baseline can be estimated by computing the temporal average of each k -space line. This baseline estimate is subtracted from both the signal distribution estimate and the lattice sampled data before filtering. The baseline is then added after the filtering step. It is argued that treating this baseline separately will avoid reconstruction errors that can otherwise be introduced in the filtering . The baseline signal does contain the by far strongest signal which could warrant special treatment since it can be estimated in this alternative way. 44 Rapid acquisition Filtering the signal distribution estimate Since the signal distribution estimate is obtained by measuring very few lines in the central parts of k -space, and reconstructed by zero-filling the outer parts, ringing artifacts may occur. Therefore, some window, typically a Hamming window, is used to reduce this ringing, though the benefits are reported to be subtle . The original k-t BLAST paper  proposes temporal low-pass filtering of the signal distribution estimate, to reduce noise with high temporal frequencies. This will also have the effect of suppressing high frequency signals, effectively lowering achieved temporal resolution. Filtering the signal distribution estimate itself will also have the unwanted side-effect of underestimating the high temporal frequencies of the aliased signal. This can be avoided by performing the temporal low-pass filtering after applying the reconstruction filter. Using signal distribution estimate twice All points in k-t space not on the lattice are reconstructed from the lattice samples using the reconstruction filter in Eq. 5.1. This includes the central k -space lines already acquired for the signal distribution estimate. Instead of using these reconstructed points, they can be substituted for the measured data, especially if the central k -space lines were acquired in an interleaved fashion during the acquisition of the lattice samples. The measured central k -space lines can also be used in the baseline estimation. This will remove any aliasing in the lower spatial frequencies of the baseline estimate. Chapter 6 Tensor field visualization Visualization of volumetric tensor fields, a three dimensional volume where each point is a tensor, is a difficult problem. The general concept of a tensor, a multilinear mapping, is often impractical to visualize. Thus, many approaches are specialized for a particular application. In the case of myocardial deformation, after eigen decomposition, three eigenvectors and corresponding eigenvalues are obtained. The eigenvectors of the strain-rate tensors represent the principal directions of instantaneous rate of shortening or lengthening. The eigenvalues indicate the rate of lengthening (positive value) or shortening (negative value). The tensor still has six degrees of freedom, making it impossible to solely rely on color coding. 6.1 Glyph visualization A common way to visualize a tensor is to use a glyph, a geometric object, that can describe the degrees of freedom of the tensor. Such a glyph can for instance consist of three arrows, representing the eigenvectors, scaled by their corresponding eigenvalues. Since the eigenvalues can be negative, this can be shown by the color of the arrow, or by making two arrows, pointing outward for positive eigenvalues or inward for negative eigenvalues, as in Figure 6.1a. A more intuitive way for the application of deformation is to visualize the tensor as an ellipsoid with the principal axes along the eigenvectors and the corresponding radii set to some basis raised to the power of the eigenvalue . This maps negative eigenvalues to a radius between 0 and 1 and positive eigenvalues to a value larger than 1. The resulting ellipsoid would be the result of deforming a unit sphere with this strain-rate for some period of time. A two-dimensional example is shown in Figure 6.1b. 46 Tensor field visualization a b Figure 6.1: Glyph visualization of deformation. Arrow glyph representation showing eigenvectors where arrowheads indicate the sign of the eigenvalues (a). In the deformed circle visualization (b), an ellipse is drawn with principal axes along the eigenvectors and corresponding radii set as a function of the eigenvalues. For visualization of strain-rate, the ellipse represents a circle after being affected by the strain-rate for some period of time. Using glyph rendering will allow description of all the degrees of freedom for a tensor. The approach is thus good for a single tensor, but becomes impractical for visualizing a whole field of tensors. Occlusion and visual cluttering makes this approach undesirable. The method suggested in Paper I is to separate visualization of a specific tensor of interest and the rest of the tensor field. The idea is to show all degrees of freedom for one tensor using glyph visualization, while some simplified global approach is used for the underlying tensor field. The tensor of interest can then be changed by navigating through the overview visualization. 6.2 Noise field filtering A popular method in vector field visualization is Line Integral Convolution (LIC) . It works by convolving a noise field along line integrals of the vector field. The convolution kernel is a smoothing kernel, typically a boxcar or Gaussian. The result, as illustrated in Figure 6.2, is a painting-like image with strokes along the vector field. The vector field is thus visualized using a scalar field, which basically contains structured white noise. The resolution of the scalar field typically needs to be higher than the resolution of the vector field, because structure with spatial extent is used to represent the vector value at a single point. One of the advantages of LIC is that it preserves continuity of the vector field. The structures are connected along the vector field. LIC has been adopted for tensor visualization by performing the convolution 6.2 Noise field filtering a 47 b Figure 6.2: Two-dimensional LIC visualization for vector field visualization (a) of the vector field in (b). Note the continuous representation and lack of directionality in the LIC visualization. sequentially for the two dominant eigenvectors . This was done with fixed convolution sizes, disregarding the degree of anisotropy, the relation between the eigenvalues, of the tensor field. The approach taken in Paper I is instead to keep the main idea of LIC, starting with a noise field and filter it according to the tensor field. The output is a scalar field, with structure representing the tensor field, that in the three dimensional case can be visualized using volume rendering. The difference with respect to LIC is that filtering is no longer performed along a line, but in a linear, planar or spherical fashion or somewhere in between, depending on the tensor field using adaptive filtering. 6.2.1 Adaptive filtering A method that uses adaptive filtering controlled by a tensor field is image enhancement [83, 84]. It is used to perform anisotropic low-pass filtering of images, avoiding filtering across edges and thereby blurring the image. In the first step, the local structure of the image is estimated into a structure tensor with the use of quadrature filters. This structure tensor contains large eigenvalues in directions of strong edges. In two dimensions, a tensor with two large eigenvalues corresponds to a point-like structure in the image, a tensor with one large eigenvalue corresponds to an edge and a tensor with two small eigenvalues corresponds to a homogeneous region 48 Tensor field visualization with no structure. In the first case, no low-pass filtering will be performed as the image contains high frequencies in all directions. In the second case, anisotropic low-pass filtering will be performed along the edge (across the eigenvector direction). In the last case, isotropic low-pass filtering is performed since there is no strong image structure. This method readily extends to higher dimensions. The filtering is performed by steerable filters. Instead of constructing a new filter for each neighborhood, a set of filters is used, consisting of one isotropic low-pass filter and several directed high-pass filters, as shown in Figure 6.3. The image is filtered separately using all filters, producing seven filter outputs in 3D. The adaptive filter output is produced by weighting the filter responses individually at each point according to the structure tensor. If the low-pass filter is combined with a high-pass filter in some direction, the result is an all-pass filter in that direction and a low-pass filter in the other directions. The high-pass components are added in directions with large eigenvalues, to preserve edges along these directions. a b c d Figure 6.3: The filter set for two-dimensional steerable anisotropic filtering, consisting of one isotropic low-pass filter (a) and three directional high-pass filters (b-d). The filters can be combined linearly into an anisotropic low-pass filter of any direction. The details of how the filter set is constructed and how the filter responses are combined are described in Paper I. In the image enhancement method, a structure tensor is estimated from the image, and the structure tensor is then used to control the filtering by steering the filtering process. In the tensor visualization approach, the tensor field is already given. The image being filtered is an initial noise image. It is beneficial to iterate the filtering process in this case, in order to be able to create curved noise structures. In image enhancement, all-pass filtering is performed in directions of high eigenvalues and low-pass filtering is performed in directions of low eigenvalues. In visualization of strain-rate tensors, the opposite is desired. This means low-pass filtering along strong eigenvalue directions, smearing the noise in these directions. The eigenvalues are therefore remapped to 6.2 Noise field filtering 49 facilitate this. A further improvement of the method described in Paper I is to make use of the sign of the eigenvalue in this mapping, similar to the mapping of eigenvalues to radii of the ellipsoid. In this way, it is possible to achieve strong low-pass filtering along positive eigenvalue directions, medium low-pass filtering along zero eigenvalue directions and all-pass filtering along negative eigenvalue directions. In Figure 6.4, this is shown for a systolic and a diastolic cardiac phase using volume rendering. Onedirectional volume preserving expansion, implying contraction in the other two directions, is thus visualized as spike-like structure in the expansion direction. Correspondingly, volume preserving one-directional contraction is shown as disc-like structure. The resulting structure is thus reminiscent of the ellipsoid. a b Figure 6.4: Tensor visualization of strain-rate tensors in the heart wall of the left ventricle in a short-axis slab in systole (a) and diastole (b). The rightventricle has been excluded for the purpose of clarity. In three dimensions, there is spike appearance in systole, depicting radial expansion, and onion layer appearance in diastole, depicting radial contraction. This method can also been used to visualize other tensor fields. One example is diffusion tensor data representing fiber structure in the brain measured with MRI. In this case, there are no negative eigenvalues, easing eigenvalue remapping. Fiber tractography, a popular method for visual- 50 Tensor field visualization izing this type of data, creates a vector field out of the tensor data by explicit tracking , but runs into problems at locations of fiber crossings where the vector model is inappropriate. An intrinsic tensor visualization approach handles this automatically. The method described in Paper I has successfully been applied on diffusion tensor data , and an output volume is shown in Figure 6.5. Figure 6.5: Volume visualization of fiber structure of the Corona Radiata in the human brain from two viewpoints. The volume is color coded according to the direction of the eigenvector corresponding to the largest eigenvalue (red = right–left, green = anterior–posterior, blue = superior–inferior). Corpus Callosum can be seen as the red structure. The green structures on top of Corpus Callosum (bottom image) are the Superior Longitudinal Fasciculus. The motor-sensory fibers can be seen as the blue structure. Chapter 7 Summary of papers This chapter summarizes the six papers included in this thesis, and my contribution to each of them. Paper I: Tensor Field Visualisation using Adaptive Filtering of Noise Fields combined with Glyph Rendering A. Sigfridsson, T. Ebbers, E. Heiberg, L. Wigström Published in Proceedings of IEEE Visualization 2002. This paper presents a method for tensor field visualization that integrates visualization of a single tensor-of-interest with an overview visualization of the complete tensor field. The idea is to use glyph rendering to show the tensor-of-interest with all degrees of freedom, while avoiding cluttering and occlusion associated with glyph rendering of tensor fields. The rest of the tensor field is then visualized using an alternative approach. A cursor can be navigated through the background to change the location of the tensor-of-interest in real time. For the background visualization, a new method inspired by line integral convolution is presented. A scalar field is created from the tensor field with the use of tensor-controlled adaptive filtering. A noise volume is used as a seed input and then iteratively filtered, creating structure in the noise revealing the directions of strong eigenvalues of the tensor field. For this paper, I implemented the filtering software and the volume rendering visualization. I also wrote the major part of the manuscript. 52 Summary of papers Paper II: Five-dimensional MRI Incorporating Simultaneous Resolution of Cardiac and Respiratory Phases for Volumetric Imaging A. Sigfridsson, J.-P. E. Kvitting, H. Knutsson, L. Wigström Published in Journal of Magnetic Resonance Imaging 2007. This paper presents a novel method for volumetric MRI acquisition temporally resolved over both cardiac and respiratory cycles simultaneously. This creates a five-dimensional data set, opening new possibilities for studying physiological effects caused by respiration on cardiac function. The method is based on a novel gating approach extended to two temporal dimensions. The acquisition is controlled in real-time by continuously estimating cardiac and respiratory phase and sampling k -space individually for each time frame. The order of k -space traversal is optimized by using a Hilbert curve which minimizes jumps in k -space. This reduces eddy currents that would cause image artifacts due to phase disruptions in k -space. For this paper, I came up with the idea, implemented the pulse sequence on the MRI scanner, the reconstruction software and the analysis software. I also wrote the major part of the manuscript. Paper III: k-t2 BLAST: Exploiting Spatiotemporal Structure in Simultaneously Cardiac and Respiratory Time-resolved Volumetric Imaging A. Sigfridsson, L. Wigström, J.-P. E. Kvitting, H. Knutsson Published in Magnetic Resonance in Medicine 2007. This paper presents an efficient way of sampling five-dimensional data based on the k-t BLAST technique, extended to two temporal dimensions. The acquisition time is reduced by sparsely sampling the k-t space on a lattice, which causes signal aliasing in the corresponding x-f space. This aliasing is suppressed by the use of a Wiener filter approach, controlled by an estimate of the signal dynamics. Using this method, an increase of spatial resolution by a factor of four is possible in half the scan time compared to full sampling in k-t space. For this paper, I came up with the idea, implemented the pulse sequence on the MRI scanner, the reconstruction software and the analysis software. I also wrote the major part of the manuscript. 53 Paper IV: Improving Temporal Fidelity in k-t BLAST MRI Reconstruction A. Sigfridsson, M. Andersson, L. Wigström, J.-P. E. Kvitting, H. Knutsson Published in Proceedings of MICCAI 2007. In k-t BLAST, signal aliasing caused by the sparse sampling is suppressed by using a Wiener filter controlled by the estimated signal dynamics. The conventional filter used is chosen to minimize the reconstruction error in a least squares sense. With high reduction factors, this results in temporal smoothing of the dynamic data, which may obscure the motion of interest. In this work, a modified reconstruction filter is proposed that preserves more of the high temporal frequencies, at the expense of slight increase in residual aliasing. Compared to the conventional k-t BLAST reconstruction, the modified filter produced images with sharper temporal delineation of the myocardial walls, which lead to more accurate estimations of wall motion. For this paper, I came up with the idea, implemented the simulation and the analysis software. I also wrote the major part of the manuscript. Paper V: Single Breath Hold Multiple Slice DENSE MRI A. Sigfridsson, H. Haraldsson, T. Ebbers, H. Knutsson, H. Sakuma In press, Magnetic Resonance in Medicine. This paper presents a technique to acquire multiple displacement encoded slices interleaved in the cardiac cycle. In particular, acquisition of three slices in a single breath hold is implemented and evaluated. The proposed method compares favorably to conventional single-slice acquisitions acquired in separate breath holds, despite the three-fold reduction in acquisition time. Being able to acquire three slices in the same breath hold reduces the risk of changes in heart rate or breath hold positions and facilitates the use of DENSE in a clinical routine protocol. For this paper, I implemented the major part of the pulse sequence on the MRI scanner, the reconstruction software and the analysis software. I also wrote the major part of the manuscript. 54 Summary of papers Paper VI: In-vivo SNR in DENSE MRI; temporal and regional effects of field strength, receiver coil sensitivity, and flip angle strategies A. Sigfridsson, H. Haraldsson, T. Ebbers, H. Knutsson, H. Sakuma Submitted manuscript. The signal in DENSE is constantly reduced during the cardiac cycle by the T1 relaxation. The signal is further attenuated due to the excitation performed for image read-out. The number and strength of excitations thus dictate the SNR behavior. The SNR is also affected by the regional coil sensitivity and the field strength used. This paper evaluates the implications of excitation patterns, receiver coils and field strength on the SNR of DENSE in-vivo. SNR was found to vary greatly on these factors. Regional coil sensitivity was found to have as much impact as a doubling of field strength from 1.5T to 3T. A commonly used excitation pattern for cine DENSE can be improved upon by more than 50% in the first half of the cardiac cycle at the cost of only slightly inferior SNR in the last 20% of the cardiac cycle. For this paper, I implemented the major part of the pulse sequence on the MRI scanner, the reconstruction software and the analysis software. I also wrote the major part of the manuscript. Chapter 8 Discussion In this thesis, new methods for imaging cardiac motion have been presented. Acquisition techniques ranging from combined cardiac and respiratory synchronization to velocity and displacement quantification in multiple slices have been described. Reconstruction from sparsely sampled data has been introduced for two temporal dimensions and an alternative reconstruction filter has been evaluated, demonstrating that least-squares minimization of the reconstruction error is not always the best choice. Furthermore, a commonly used flip angle strategy, optimized for maximum constant SNR was found to yield considerably lower SNR than a fixed flip angle for the greater part of the cardiac cycle. Visualization approaches for strain-rate tensor fields have also been presented. Below, related concepts and topics of interest are discussed. 8.1 Multidimensional imaging This thesis presents imaging methods for high dimensional data. Myocardial deformation was represented as a tensor field with six degrees of freedom in every point in a time-resolved volume. This corresponds to an inner dimension of six and an outer dimension of four. Imaging of anatomical structures in the five-dimensional approach, resolving the cardiac and respiratory cycles in a volumetric acquisition, has an outer dimension of five. In Papers II, III and IV, only scalar image data were acquired, i.e. having an inner dimensionality of one. Routine clinical work as of today, on the other hand, often use traditional methods in lower dimensions. Echocardiography measures ultrasound reflectance or a one-directional Doppler shift in each sample point, corresponding to an inner dimension of one. This is performed along 56 Discussion one spatial line over time in M-mode echocardiography. Two-dimensional echocardiography extends this with one additional outer spatial dimension. There is no question that imaging in higher dimensions is more difficult and time-consuming, and thus more expensive. One may ask oneself what the extra dimensions add to what is used in clinical practice today. When it comes to physiological understanding, especially of the more complex interactions such as interventricular coupling or local myocardial deformation, directions and variations of motion is not known beforehand. Even if measuring motion in only one direction imposes implicit assumptions this can, in the hands of a skillful operator, be reasonable for some applications. For research purposes, however, objectivity is important, and the goal is to reduce sensitivity to operator dependent acquisition, slice positioning, slice misregistration and angular error in velocity measurements. MRI studies have stressed the need of a three-dimensional characterization of the shape and curvature of the septum . More comprehensive physiological understanding can lead to better assumptions regarding motion directionality and slice orientation sensitivity. This can then be transferred into lower dimensional methods, and to clinical routine. Papers V and VI describe methods which acquire myocardial deformation in a single breath hold, which is particularly suited for clinical routine examination. These describe acquisition of strain tensors with three degrees of freedom on one or several single- or multi-phase slices. In practice, the tensor is often reduced to only the eigenvalues for easier interpretation. This is an implementation of the idea of multidimensional acquisition with data reduction in the final analysis step. 8.2 Costs of sparse sampling The reduction of acquisition time in k-t BLAST comes from sampling the k-t space less densely. There are definitively drawbacks of the sparse sampling of data, such as increased noise and reduced temporal fidelity. 8.2.1 Noise It is in the nature of MRI that a shorter acquisition time means more noise. One may visualize this as aliased copies of noise; i.e. that the signal in x-f space is not really limited to the localized signal bearing parts, but there is a wideband noise component as well. If there are fewer sampling points in k-t space, more noise will interfere with the desired signal. The high reduction factors typically used in k-t BLAST means that very few data points are acquired. This will have effects on noise. In the application 8.2 Costs of sparse sampling 57 of cardiac imaging, this extra noise is usually accepted in order to reduce imaging time or increase resolution. Since voxel sizes in typical cardiac examinations are quite large, there is quite a lot of signal to begin with. Also, k-t BLAST carries an implicit Wiener filtering reducing the noise temporally. Nevertheless, spatial filtering and especially spatiotemporal filtering, along the lines of image enhancement [83, 84], might prove useful to reduce noise when high spatial resolutions are approached. In the case of five-dimensional imaging, edges have a lot of structure that can be exploited in noise reduction. 8.2.2 Temporal fidelity The aim of the k-t BLAST method is to reduce imaging time without sacrificing image quality. The reconstruction error occurs where the signals overlap in x-f space. Overlapping mostly occurs between high temporal frequencies of one spatial position and low temporal frequencies of a different spatial position. The reconstruction error is most prominent in high temporal frequencies. This is because the signal energy in the lower temporal frequencies is much stronger than in the high temporal frequencies and the optimal linear reconstruction with such overlap is to attenuate the higher frequencies. The result of this is a lower effective temporal bandwidth, i.e. temporal blurring. This mechanism of alias overlap attenuation, suppressing the high temporal frequencies to retain the lower frequency content, will provide static images from k-t BLAST of very high quality. The aliasing which would arise if zero-filling reconstruction was performed will be minimized. The loss of temporal bandwidth is only fully exposed in the temporal dimension, and since this dimension is omitted in static images, a false impression of retained fidelity is presented. It is therefore important to study the temporal dimension when using k-t BLAST. This can either be done with the use of animation or by using M-mode type visualization, where the temporal dimension is presented spatially. It should be noted that the loss of temporal bandwidth is not equivalent to sampling with a lower frequency. Sometimes, high temporal bandwidth is used to suppress motion artifacts, not to resolve a particular temporal event. The temporal blurring occurring in k-t BLAST is caused by filtering out the high frequency content, thereby avoiding aliasing onto the lower frequencies. In this case, the loss of effective temporal bandwidth is not a loss of relevant image information. With the use of short-TR pulse sequences, such as balanced SSFP, it is possible to trade scan time for improved temporal resolution by reducing 58 Discussion k -space segmentation. By further applying k-t BLAST, the original scan time can be restored. In this way, the original temporal bandwidth should at least be preserved, but hopefully increased, because the bandwidth will be shared between different spatial positions. With a reduction factor of N, N different spatial positions will share an N-fold bandwidth. If some spatial positions have low bandwidth, which is very common in cardiac imaging, more sampling bandwidth is available to the other positions. What remains to be studied is the practical gain achievable and the cost of acquiring the signal distribution estimate and its effect on the accuracy of the Wiener reconstruction filter. With two temporal dimensions, several parts of the k-t BLAST approach can be controlled to act differently in each dimension. One may, for instance, adjust lattice optimization in order to preserve temporal fidelity in one temporal dimension at the cost of increasing temporal blurring in the other dimension. Some applications desire high cardiac phase fidelity, but might accept a larger imaging window in the respiratory cycle, if k-t2 BLAST is used as a respiratory artifact reduction method. Other applications desire high respiratory phase fidelity, but can accept temporal blurring in cardiac phase dimension if the imaging is concentrated on the slow-varying cardiac phases. 8.3 Future work As with all new methods, it is vital to put the new tools to a test. Only by applying the new techniques can we start to validate and evaluate the methods. Where validation is infeasible because of lacking control methods, predictive value and similar measures may still be obtainable. Also, by evaluating the methods, weaknesses and points of improvement may be identified. 8.3.1 Strain and strain-rate estimation The estimation of the strain-rate tensor involves computing differentials, which in practice often is implemented by finite differencing. This means that the process is highly sensitive to noise and image artifacts. This imposes higher demands on the image quality in these applications. Adaptive differentiation, for example by using normalized convolution [88, 89] may be useful in this context. 8.3 Future work 8.3.2 59 Adapting the DENSE acquisition to the application As was seen in Paper VI, using appropriate acquisition parameters and hardware is essential when it comes to achieving sufficient SNR in DENSE. This has to be done for each application, since the stimulated echo signal is affected considerably by the acquisition itself. 8.3.3 Optimizing reduction factor versus temporal fidelity When using k-t BLAST approaches, the effect of temporal blurring needs to be considered. A large reduction factor is favorable for reducing scan time or increasing spatiotemporal resolution, but also increases risk of loosing important temporal information. The appropriate reduction factor needs to be optimized for the individual applications. 8.3.4 Using k-t2 BLAST for respiratory gating k − t2 BLAST may be used as an alternative to respiratory gating. Instead of acquiring data only in the end expiration period of the respiratory cycle, data can be acquired continuously, resolving the respiratory cycle. Compared to respiratory gating, where roughly 50% of the cardiac cycle can provide adequate data , resolving the whole respiratory cycle requires better temporal resolution in the inspiratory phases. The resulting increase in scan time can be avoided if k − t2 BLAST is used for scan time reduction. One may then after reconstruction select just one respiratory time frame to simulate a respiratory gated acquisition. An advantage is that this removes the necessity of prior knowledge of the optimal respiratory phase. 8.3.5 Acquisition of velocity data using k-t2 BLAST Two-dimensional phase-contrast MRI measurements resolving both cardiac and respiratory cycles has been presented previously  and shows promise of measuring important respiratory variations in blood flow. To this date, volumetric velocity acquisition resolved over both cardiac and respiratory cycles has not been performed. Five-dimensional acquisition in the form presented in Paper II is too time consuming to add velocity measurements. Acquisition time shortening, along the lines of k-t2 BLAST presented in Paper III, is a prerequisite for five-dimensional velocity measurements. Phasecontrast MRI has been combined with k-t BLAST , and should be extendable to k-t2 BLAST as well. Not only will these measurements be free of respiratory motion artifacts, as described in Section 8.3.4, but it will also 60 Discussion enable the studies of different flow patterns in different phases of the respiratory cycle. The right ventricle has highly respiratory dependent flow, and its complex shape is best described using a volumetric acquisition. 8.4 Potential impact The tensor field visualization presented in Paper I may help investigators in the study of local myocardial deformation using fully three-dimensional measurements. This may in turn be used to obtain new information as to which directions and locations of deformations are important for diagnosis or follow-up of certain diseases and treatments. Volumetric imaging resolving both cardiac and respiratory cycles simultaneously offers a completely new type of data. Study or quantification of shape and ventricular volumes over the course of the respiratory cycle may offer new means for physiological description of interventricular coupling. The method presented in Paper II for measuring such a data set has the potential to describe these phenomena. Possibility to shorten acquisition time or increase resolution in cardiac imaging is always desired. The method presented in Paper III offers a way to not only use temporal correlation in the cardiac cycle, but also in the combined two-dimensional space of cardiac and respiratory cycles. This has the potential of reducing reconstruction error or allowing larger reduction factors. The method was demonstrated for volumetric imaging, but may also be applied to single slice imaging for a far broader range of applications. The alternative k-t BLAST reconstruction filter presented in Paper IV was shown to improve depiction of the myocardial wall dynamics, both visually and quantitatively. This shows that minimizing the L2 reconstruction error, as is done in conventional k-t BLAST, is not always the best choice. Furthermore, this suggests that new reconstruction methods may improve the results even from already acquired raw data. Paper V presented a technique to acquire multiple displacement encoded slices in a single breath hold. Being able to image myocardial strain in the whole left ventricle in a single breath hold is an important step in incorporating DENSE in the clinical routine; not only is the examination time reduced, but also patient cooperation can be improved by the reduced breath hold burden. In Paper IV, it was found that the SNR in DENSE is highly dependent on the choice of acquisition parameters and imaging hardware. Particularly, a popular choice of flip angle strategy was found to be a poor choice for imaging systolic myocardial function. 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