The Development and Application of a Simulation System for Diffusion

Total Page:16

File Type:pdf, Size:1020Kb

The Development and Application of a Simulation System for Diffusion The development and application of a simulation system for diffusion-weighted MRI Mark Graham A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy of University College London. Centre for Medical Image Computing & Department of Computer Science University College London April 22, 2018 2 3 I, Mark Graham, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the work. Abstract Diffusion-weighted MRI (DW-MRI) is a powerful, non-invasive imaging tech- nique that allows us to infer the structure of biological tissue. It is particularly well suited to the brain, and is used by clinicians and researchers studying its structure in health and disease. High quality data is required to accurately characterise tissue structure with DW-MRI. Obtaining such data requires the careful optimisation of the image acquisition and processing pipeline, in order to maximise image quality and minimise artefacts. This thesis extends an existing MRI simulator to create a simulation sys- tem capable of producing realistic DW-MR data, with artefacts, and applies it to improve the acquisition and processing of such data. The simulator is applied in three main ways. Firstly, a novel framework for evaluating post- processing techniques is proposed and applied to assess commonly used strate- gies for the correction of motion, eddy-current and susceptibility artefacts. Secondly, it is used to explore the often overlooked susceptibility-movement interaction. It is demonstrated that this adversely impacts analysis of DW- MRI data, and a simple modification to the acquisition scheme is suggested to mitigate its impact. Finally, the simulation is applied to develop a new tool to perform automatic quality control. Simulated data is used to train a classifier to detect movement artefacts in data, with performance approaching that of a classifier trained on real data whilst requiring much less manually-labelled training data. It is hoped that both the findings in this thesis and the simulation tool itself will benefit the DW-MRI community. To this end, the tool is made freely 6 Abstract available online to aid the development and validation of methods for acquiring and processing DW-MRI data. Acknowledgements I owe thanks to my supervisors, Gary and Ivana. Gary’s time, attention to detail, and high standards greatly improved the quality of my PhD work. Ivana was an invaluable source of knowledge on MR physics and POSSUM, and provided a great amount of advice, academic and otherwise, throughout my time at UCL. I enjoyed a fruitful collaboration with Jesper and MJ from FMRIB, Oxford - thanks for your enthusiasm for the simulator, and the chance to contribute to eddy’s development. MIG provided a great environment for research. I’m glad to have been part of a research group with so many students: thanks for all discussions, cookies and conference fun. I was also fortunate to meet Alex, Eliza, Felix, and Zach: thanks for all the laughter, and ensuring I never took anything too seriously. I am grateful to all my friends and family. Thank you Mum, Dad, and Helen, for all your love, support, and tolerating me never wanting to talk about my thesis! So many friends have made the last four years so fun, but special mention is owed to the residents of the Deptford Commune: Emily, George, Javal, Sinead, and Prev. Lastly, to Lindsay. You’ve provided love, encouragement, and the ideal location for me to start writing this thesis: a hammock in a house in Zanzibar, jealously guarded by a very silly cat. Contents 1 Introduction 17 1.1 Motivation . 17 1.2 Problem statement . 19 1.3 Project aims and scope . 19 1.4 Thesis overview and contributions . 19 2 Background 23 2.1 Magnetic resonance imaging . 23 2.1.1 Protons in a magnetic field . 24 2.1.2 Detection of the magnetization . 25 2.1.3 Spatial encoding . 26 2.1.4 Image formation and contrast . 30 2.2 Diffusion MRI . 34 2.2.1 Diffusion . 35 2.2.2 Measuring diffusion with MRI . 37 2.2.3 Models of diffusion . 40 2.3 Image artefacts . 43 2.4 Simulation systems . 48 2.4.1 MRI . 48 2.4.2 Diffusion MRI . 51 2.4.3 Critical analysis . 53 3 A simulation system for diffusion-weighted MRI 55 10 Contents 3.1 Overview . 55 3.1.1 Research dissemination . 55 3.2 Introduction . 56 3.3 Methods . 57 3.3.1 Framework overview . 57 3.3.2 Producing DWIs . 58 3.3.3 Incorporating artefacts . 61 3.3.4 Ground truth displacement fields . 62 3.4 Simulator demonstration . 64 3.4.1 Image contrast . 64 3.4.2 Eddy-current artefacts . 65 3.5 Discussion . 65 3.6 Conclusions . 70 4 Application I: assessing motion and eddy-current correction techniques 71 4.1 Overview . 71 4.1.1 Research dissemination . 71 4.2 Introduction . 72 4.3 Experiments and Results . 73 4.3.1 Comparison of correction methods . 73 4.3.2 Evaluation of eddy ..................... 76 4.3.3 Impact of correction on microstructure estimation . 80 4.4 Discussion . 85 4.5 Conclusions . 92 5 Simulating the spin-echo 93 5.1 Overview . 93 5.1.1 Research dissemination . 93 5.2 Introduction . 93 5.3 Methods . 94 Contents 11 5.3.1 Implementation of spin-echo in POSSUM . 94 5.3.2 Incorporation of POSSUM into the DW-MR framework . 97 5.4 Validation . 98 5.5 Discussion . 100 6 Application II: investigating the susceptibility artefact 103 6.1 Overview . 103 6.1.1 Research dissemination . 103 6.2 Introduction . 104 6.3 Methods . 107 6.3.1 Assessment of existing techniques using simulated data . 107 6.3.2 Assessment of existing techniques using real data . 111 6.3.3 Assessment of the susceptibility-movement interaction . 113 6.4 Results . 114 6.4.1 Assessment of techniques with simulated data . 114 6.4.2 Assessment of techniques with real data . 122 6.4.3 Interaction between susceptibility and movement . 122 6.5 Discussion . 125 6.6 Conclusions . 132 7 Application III: a tool for automated quality control 135 7.1 Overview . 135 7.1.1 Research dissemination . 135 7.2 Introduction . 136 7.3 Methods . 138 7.3.1 Data . 138 7.3.2 Classifier . 143 7.4 Results . 145 7.5 Discussion . 147 7.6 Conclusion . 151 12 Contents 8 Conclusions 153 8.1 Summary . 153 8.2 Future directions . 154 8.2.1 Simulator improvements . 154 8.2.2 Simulator object improvements . 155 8.2.3 Further applications . 156 Appendices 158 A Publications 159 Bibliography 161 List of Figures 2.1 Motion of M~ during an 90° pulse in the rotating and lab frames. 27 2.2 The gradient-echo pulse sequence. 29 2.3 Image formation in the case of discrete, finite sampling. 32 2.4 Image contrast as a function of TR and TE............. 35 2.5 Illustration of diffusion in a cylinder. 37 2.6 Variation in contrast with b-value. 38 2.7 Schematic of the Stejskal-Tanner pulse sequence . 40 2.8 A single-shot EPI scheme. 41 3.1 The pipeline for simulating DWIs. 59 3.2 Comparison of real and simulated data. 66 3.3 Comparison of our simulated data to state-of-the-art. 67 3.4 Demonstration of DWIs simulated with EC artefacts. 68 4.1 Mean error in displacement field across the brain. 77 4.2 Spatial errors in displacement field. 78 4.3 Correction errors on real data. 79 4.4 Errors after correction with eddy.................. 81 4.5 Errors after correction with eddy.................. 82 4.6 FA maps for corrected data. 85 4.7 NODDI maps for corrected data. 86 4.8 Parameter maps resulting from fits to real data. 87 4.9 Model residuals after correction. 88 4.10 Mean error field over the brain. 89 14 List of Figures 4.11 Spatial plots of the mean error field. 90 5.1 Flowchart describing the POSSUM simulator. 95 5.2 The new framework for simulating DW-MR data. 98 5.3 Comparison of simulation with theory for a spin-echo experiment. 99 5.4 Comparison between simulated GE-EPI and SE-EPI images. 99 5.5 Simulations both with and without a linear B0 gradient across the FOV. 100 5.6 Comparison between GE-EPI and SE-EPI images with a sus- ceptibility field. 101 6.1 Simulated DWIs. 112 6.2 Displacement field errors. 115 6.3 Errors in image intensity. 118 6.4 Errors in FA metrics. 120 6.5 AP-LR comparison on simulated data. 123 6.6 AP-LR comparison on real data. 124 6.7 Susceptibility-movement interaction. 126 6.8 Errors introduced when failing to account for the susceptibility- movement interaction. 127 6.9 Error distribution. 128 7.1 Example of the intra-volume movement artefact. 139 7.2 Example simulated motion trace. 141 7.3 Real and simulated data with QC labels. 146 7.4 Precision-recall curve for both classifiers. 147 7.5 Sample classifier results for images in the test-set. 148 7.6 Quantitative classifier results on the test set. ..
Recommended publications
  • A Beginner's Guide to Bloch Equation Simulations of Magnetic
    A Beginner’s Guide to Bloch Equation Simulations of Magnetic Resonance Imaging Sequences ML Lauzon, PhD Depts of Radiology and Clinical Neurosciences, University of Calgary, Calgary, AB, Canada Seaman Family MR Research Centre, Calgary, AB, Canada [email protected] ABSTRACT Nuclear magnetic resonance (NMR) concepts are rooted in quantum mechanics, but MR imaging principles are well described and more easily grasped using classical ideas and formalisms such as Larmor precession and the phenomenological Bloch equations. Many textbooKs provide in-depth descriptions and derivations of the various concepts. Still, carrying out numerical Bloch equation simulations of the signal evolution can oftentimes supplement and enrich one’s understanding. And though it may appear intimidating at first, performing these simulations is within the realm of every imager. The primary objective herein is to provide novice MR users with the necessary and basic conceptual, algorithmic and computational tools to confidently write their own simulator. A brief background of the idealized MR imaging process, its concepts and the pulse sequence diagram are first provided. Thereafter, two regimes of Bloch equation simulations are presented, the first which has no radio frequency (RF) pulses, and the second in which RF pulses are applied. For the first regime, analytical solutions are given, whereas for the second regime, an overview of the computationally efficient, but often overlooked, Rodrigues’ rotation formula is given. Lastly, various simulation conditions of interest and example code snippets are given and discussed to help demonstrate how straightforward and easy performing MR simulations can be. INTRODUCTION Magnetic resonance (MR) imaging is one of the most versatile imaging modalities used clinically.
    [Show full text]
  • 4 Nuclear Magnetic Resonance
    Chapter 4, page 1 4 Nuclear Magnetic Resonance Pieter Zeeman observed in 1896 the splitting of optical spectral lines in the field of an electromagnet. Since then, the splitting of energy levels proportional to an external magnetic field has been called the "Zeeman effect". The "Zeeman resonance effect" causes magnetic resonances which are classified under radio frequency spectroscopy (rf spectroscopy). In these resonances, the transitions between two branches of a single energy level split in an external magnetic field are measured in the megahertz and gigahertz range. In 1944, Jevgeni Konstantinovitch Savoiski discovered electron paramagnetic resonance. Shortly thereafter in 1945, nuclear magnetic resonance was demonstrated almost simultaneously in Boston by Edward Mills Purcell and in Stanford by Felix Bloch. Nuclear magnetic resonance was sometimes called nuclear induction or paramagnetic nuclear resonance. It is generally abbreviated to NMR. So as not to scare prospective patients in medicine, reference to the "nuclear" character of NMR is dropped and the magnetic resonance based imaging systems (scanner) found in hospitals are simply referred to as "magnetic resonance imaging" (MRI). 4.1 The Nuclear Resonance Effect Many atomic nuclei have spin, characterized by the nuclear spin quantum number I. The absolute value of the spin angular momentum is L =+h II(1). (4.01) The component in the direction of an applied field is Lz = Iz h ≡ m h. (4.02) The external field is usually defined along the z-direction. The magnetic quantum number is symbolized by Iz or m and can have 2I +1 values: Iz ≡ m = −I, −I+1, ..., I−1, I.
    [Show full text]
  • The Classical Bloch Equations
    The classical Bloch equations Martin Frimmer and Lukas Novotny ETH Zurich,€ Photonics Laboratory, 8093 Zurich,€ Switzerland (www.photonics.ethz.ch) (Received 13 October 2013; accepted 7 May 2014) Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes, and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled- harmonic-oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times (T1 and T2) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories. VC 2014 American Association of Physics Teachers. [http://dx.doi.org/10.1119/1.4878621] I. INTRODUCTION offers students an intuitive entry into the field and prepares them with the basic concepts of quantum coherent control. The harmonic oscillator is arguably the most fundamental building block at the core of our understanding of both clas- sical and quantum physics. Interestingly, a host of phenom- II. THE MECHANICAL ATOM ena originally encountered in quantum mechanics and A. Equations of motion initially thought to be of purely quantum-mechanical nature have been successfully modelled using coupled classical har- Throughout this paper, we consider two oscillators, as monic oscillators.
    [Show full text]
  • MRI Notes 1: Page 1
    Noll (2006) MRI Notes 1: page 1 Notes on MRI, Part 1 Overview Magnetic resonance imaging (MRI) – Imaging of magnetic moments that result from the quantum mechanical property of nuclear spin. The average behavior of many spins results in a net magnetization of the tissue. The spins possess a natural frequency that is proportional to the magnetic field. This is called the Larmor relationship: ω = γB Any magnetization that is transverse (perpendicular) to an applied magnetic field B will precess around that B field at the Larmor frequency. In MRI there are 3 kinds of magnetic fields: 1. B0 – the main magnetic field 2. B1 – an RF field that excites the spins 3. Gx, Gy, Gz – the gradient fields that provide localization The major steps in a 1D MRI experiment are (we’ll do 2 and 3 acquisitions later): 1. Object to be imaged is placed into the main field, B0. Subsequently, the object develops a distribution of magnetization, m0(x,y,z), that is to be imaged. This magnetization is aligned with B0 (in the z-direction). 2. A rotating RF magnetic field, B1, is applied to tip the magnetization into the plane that is transverse to B0. While in this plane, the magnetization precesses about the main field at a Noll (2006) MRI Notes 1: page 2 frequency proportional to the strength of the main field (ω = γB). This precessing magnetization creates a voltage in a receive coil, which is acquired for subsequent processing. z v(t) v(t) B M t y ω0 x 3. Gradient magnetic fields are applied to set-up a one-to-one correspondence between spatial position and frequency.
    [Show full text]
  • FELIX BLOCH October 23, 1905-September 10, 1983
    NATIONAL ACADEMY OF SCIENCES F E L I X B L O C H 1905—1983 A Biographical Memoir by RO BE R T H OFSTADTER Any opinions expressed in this memoir are those of the author(s) and do not necessarily reflect the views of the National Academy of Sciences. Biographical Memoir COPYRIGHT 1994 NATIONAL ACADEMY OF SCIENCES WASHINGTON D.C. FELIX BLOCH October 23, 1905-September 10, 1983 BY ROBERT HOFSTADTER ELIX BLOCH was a historic figure in the development of Fphysics in the twentieth century. He was one among the great innovators who first showed that quantum me- chanics was a valid instrument for understanding many physi- cal phenomena for which there had been no previous ex- planation. Among many contributions were his pioneering efforts in the quantum theory of metals and solids, which resulted in what are called "Bloch Waves" or "Bloch States" and, later, "Bloch Walls," which separate magnetic domains in ferromagnetic materials. His name is associated with the famous Bethe-Bloch formula, which describes the stopping of charged particles in matter. The theory of "Spin Waves" was also developed by Bloch. His early work on the mag- netic scattering of neutrons led to his famous experiment with Alvarez that determined the magnetic moment of the neutron. In carrying out this resonance experiment, Bloch realized that magnetic moments of nuclei in general could be measured by resonance methods. This idea led to the discovery of nuclear magnetic resonance, which Bloch origi- nally called nuclear induction. For this and the simulta- neous and independent work of E.
    [Show full text]
  • Transient Chaos in Fractional Bloch Equations Sachin Bhalekar A, Varsha Daftardar-Gejji B, Dumitru Baleanu C,D,∗, Richard Magin E
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Computers and Mathematics with Applications 64 (2012) 3367–3376 Contents lists available at SciVerse ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Transient chaos in fractional Bloch equations Sachin Bhalekar a, Varsha Daftardar-Gejji b, Dumitru Baleanu c,d,∗, Richard Magin e a Department of Mathematics, Shivaji University, Kolhapur - 416004, India b Department of Mathematics, University of Pune, Pune - 411007, India c Department of Mathematics and Computer Science, Faculty of Arts and Sciences, Cankaya University, 06530, Ankara, Turkey d Institute of Space Sciences, P.O.Box, MG-23, R 76900, Magurele-Bucharest, Romania e Department of Bioengineering, University of Illinois, 851 S. Morgan St., Chicago, 60607, USA article info a b s t r a c t Keywords: The Bloch equation provides the fundamental description of nuclear magnetic resonance Fractional calculus (NMR) and relaxation (T1 and T2). This equation is the basis for both NMR spectroscopy and Bloch equation magnetic resonance imaging (MRI). The fractional-order Bloch equation is a generalization Chaos of the integer-order equation that interrelates the precession of the x, y and z components of magnetization with time- and space-dependent relaxation. In this paper we examine transient chaos in a non-linear version of the Bloch equation that includes both fractional derivatives and a model of radiation damping. Recent studies of spin turbulence in the integer-order Bloch equation suggest that perturbations of the magnetization may involve a fading power law form of system memory, which is concisely embedded in the order of the fractional derivative.
    [Show full text]
  • 14 the Classical Description of NMR
    14 The Classical Description of NMR 14.1 Introduction In the previous section we showed how the mysterious quantum mechanical property of spin possessed by the hydrogen nuclei (e.g., protons) in tissue water resulted in a macroscopic net or bulk magnetization M that is small but measureable when a large number of spins were placed in a large, static magnetic field Bo. This magnetization points in the same direction as B , which is usually defined to be the zˆ direction, and there is no component in the x y plane, o − because these “transverse” components of the many spins point in random directions, and thus average to zero. Moreover, the way this magnetization interacts with an arbitrary, time-varying external magnetic field B(t), its so-called equation of motion, can be described in completely classical terms, i.e., one need not know quantum mechanics. The equation of motion for the magnetization is the simple equation dM = γM B (14.1) dt ⇥ where is the cross product discussed in Section 3.14. From our discussion of the cross product ⇥ in Section ??, we can write this in the form of a matrix equation (dropping the explicit time dependence for simplicity): dM = ⌦M (14.2) dt where M 0 B B x − z y M = M , ⌦ = γ B 0 B (14.3) 0 y1 0 z − x1 Mz By Bx 0 @ A @− A It will be helpful to recall that the matrix form of the cross product is formed from the dot product of B with the component matrices Eqns ??- ?? (called the infinitesimal generators) representing the cross product: ⌦ = γBxAx + γByAy + γBzAz (14.4) This expression is very useful for investigating di↵erent components of the applied magnetic fields.
    [Show full text]
  • Lecture #2 Review of Classical MR
    Lecture #2! Review of Classical MR • Topics – Nuclear magnetic moments – Bloch Equations – Imaging Equation – Extensions • Handouts and Reading assignments – van de Ven: Chapters 1.1-1.9 – de Graaf, Chapters 1, 4, 5, 10 (optional). – Bloch, “Nuclear Induction”, Phys Rev, 70:460-474, 1946 – Historical Notes • Lauterbur, “Image Formation by Induced Local Interactions: Examples Employing Nuclear Magnetic Resonance”, Nature 242:190-191, 1973. • Mansfield and Grannell, “NMR ‘Diffraction’ in solids?”, J. Phys. C: Solid State Phys., 6:L422-L426, 1973. 1 Spin • Protons (as well as electrons and neutrons) possess intrinsic angular momentum called “spin” • Spin gives rise to a magnetic dipole moment • Useful (though not entirely accurate) to think of a proton as a spinning or rotating charge generating a current, which, in turn, produces a magnetic moment. µ 2 Nuclear Magnetic Moment • Consider a point charge in circular motion: radius r velocity v charge e • From EM theory: in the far field a current loop looks just like a magnetic dipole with magnetic moment µ µ = current • loop area ev 2 e µ = ⋅πr = mvr 2πr 2m angular momentum L gyromagnetic ratio γ • Thus µ⇥ = γL⇥ 3 Gyromagnetic Ratio • γ often expressed as gµb e γ = where µb = and = spin g factor 2m g g ≅ 2 (electrons) Planck’s constant/2 Bohr magneton π (m = electron mass) • For protons gµn e γ = where µn = and g≅ 5.6 2mp γ nuclear magneton = 42.58 MHz/T 2π γ e Important for ESR, NMR Note, for electron spin: = 658 contrast agents, etc γ K. Zavoisky 4 Nuclear Spin in a Magnetic Field • In a uniform magnetic field, a magnetic dipole will experience a torque τ Example: compass ⇥ = µ⇥ B⇥ × • Potential energy given by: angle between E = µ⇥ B⇥ = µB cos θ µ and B − · − Classically, energy can take on any value between ± µB 5 Equation of Motion • Newton’s Law: dL⇥ = ⇥ dt • Combining previous equations: dµ⇥ = ⇥µ B⇥ dt × 6 Physical Picture: ! Single Spin in a Uniform Magnetic Field dµ⇥ = ⇥µ B⇥ Note: µ = constant dt × | | B = B0zˆ z Precession frequency µ θ ω 0 ≡ γB0 Note: Some texts use ω0 = -γB0.
    [Show full text]
  • Week 1: Introduction to the Basics: Bloch Equations
    Class Notes for BCH 6746 (Spring, 2000) Structural Biology: Theory and Applications of NMR Spectroscopy Arthur S. Edison ([email protected]) Assistant Professor Biochemistry & Molecular Biology University of Florida Week 1: Introduction to the basics: Bloch equations References: Most NMR books. These notes were constructed from combinations of: “Protein NMR spectroscopy: Principles and Practice” Cavanagh, Fairbrother, Palmer, and Skelton, Academic Press, (1996). “Biomolecular NMR Spectroscopy” Evans, Oxford University Press, (1995). “Principles of Nuclear Magnetic Resonance in One and Two Dimensions” Ernst, Bodenhausen, and Wokaun, Oxford (1987). Nuclear spin and angular momentum Some atomic nuclei have an intrinsic property called “spin”. This was first demonstrated in 1922 by the Stern-Gerlach experiment, in which a beam of silver atoms were passed through a magnet field and split into two beams. These two beams represent the two states, a and b, of the silver (spin 1/2) nuclei. The nuclear spin has an intrinsic angular momentum, a vector that is represented by the symbol I (vectors will be in bold). Vectors have 3 orientations (x, y, and z) and a length. However, the Heisenberg Uncertainty Principle tells that we can only know one orientation and the length simultaneously. By convention in NMR, we say that we know the z-orientation of the angular momentum of a nucleus in a magnetic field. The square of the magnitude of I is given by 2 2 I = h I(I +1) (1-1) where I is the spin quantum number (e.g. 1/2) and h is Planck’s constant divided by 2p.
    [Show full text]
  • Lecture 13 — February 25, 2021 Prof
    MATH 262/CME 372: Applied Fourier Analysis and Winter 2021 Elements of Modern Signal Processing Lecture 13 | February 25, 2021 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long, Edited by E. Bates 1 Outline Agenda: Magnetic resonance imaging (MRI) 1. Bird's eye view: physics at play, MRI machines 2. Nuclear magnetic resonance (NMR) 3. Bloch phenomenological equations 4. Relaxation times A nice introductory video to MRI can be found here http://www.youtube.com/watch?v=Ok9ILIYzmaY Last Time: We surveyed two techniques for efficiently computing non-uniform discrete Fourier transforms. We considered two distinct tasks: Type I : evaluating the Fourier transform at uniform frequencies from non-uniform data; and Type II : evaluating the transform at non-uniform frequen- cies from uniform data. Finally, we mentioned that Taylor approximations could be employed to simultaneously consider non-uniform grids in both domains, the so-called Type III problem. 2 Magnetic resonance imaging After studying computerized tomography (CT) and its connections to the Fourier transform, we now turn to discussing the principles behind magnetic resonance imaging (MRI). As we saw with X-ray tomography, the general idea of medical imaging is to measure the outcome of an interaction between the human body and a known energy source. In MRI, one measures the response of water molecules in the human body when subjected to low-energy electromagnetic pulses against a strong magnetic field. This makes MR an attractive imaging modality to practitioners, as there is no evidence that exposure to strong magnetic fields is harmful. Of course, in order to understand how electromagnetic pulses ultimately lead to an image, we need to construct a mathematical model of the relevant physics.
    [Show full text]
  • A New Adjustment of Laplace Transform for Fractional Bloch Equation in NMR Flow
    Available at Applications and Applied http://pvamu.edu/aam Mathematics: Appl. Appl. Math. ISSN: 1932-9466 An International Journal (AAM) Vol. 9, Issue 1 (June 2014), pp. 201-216 A New Adjustment of Laplace Transform for Fractional Bloch Equation in NMR Flow Sunil Kumar Department of Mathematics National Institute of Technology Jamshedpur, 801 014, Jharkhand, India [email protected]; [email protected] Devendra Kumar Department of Mathematics Jagan Nath Gupta Institute of Engineering &Technology Jaipur- 302 022, Rajasthan, India [email protected] U. S. Mahabaleshwar Government First Grade College for Women Hassan- 573 201 Karnataka, India [email protected] Abstract This work purpose suggest a new analytical technique called the fractional homotopy analysis transform method (FHATM) for solving time fractional Bloch NMR (nuclear magnetic resonance) flow equations, which are a set of macroscopic equations that are used for modeling nuclear magnetization as a function of time. The true beauty of this article is the coupling of the homotopy analysis method and the Laplace transform method for systems of fractional differential equations. The solutions obtained by the proposed method indicate that the approach is easy to implement and computationally very attractive. Keywords: Bloch equations; Laplace transform method; approximate solution; new fractional homotopy analysis transform method AMS-MSC 2010 No.: 26A33, 34A08, 34A34, 60G22 201 202 Sunil Kumar et al. 1. Introduction In recent years, considerable interest in fractional differential equations has been stimulated by their numerous applications in the areas of physics and engineering [West (2003)]. In past years, differential equations involving derivatives of non-integer order provided adequate models for various physical phenomena [Podlubny (1999)].
    [Show full text]
  • Modified-Bloch Equation Based on Fractal Derivative for Analyzing PFG Anomalous Diffusion
    Modified-Bloch Equation Based on Fractal Derivative for Analyzing PFG Anomalous Diffusion Guoxing Lin* Carlson School of Chemistry and Biochemistry, Clark University, Worcester, MA 01610 September 2017 ABSTRACT: A modified-Bloch equation based on the fractal derivative is proposed to analyze pulsed field gradient (PFG) anomalous diffusion. Anomalous diffusion exists in many systems such as in polymer or biological systems. PFG anomalous diffusion could be analyzed based on the fractal derivative or the fractional derivative. Compared to the fractional derivative, the fractal derivative is simpler, and it is faster in numerical evaluations. In this paper, the fractal derivative is employed to build the modified-Bloch equation that is a fundamental method to describe the spin magnetization evolution affected by fractional diffusion, Larmor precession, and relaxation. An equivalent form of the fractal derivative is proposed to convert the fractional diffusion equation, which can then be combined with the precession and relaxation equations to get the modified- Bloch equation. This modified-Bloch equation yields a general PFG signal attenuation expression that includes the finite gradient pulse width (FGPW) effect, namely, the signal attenuation during field gradient pulse. The FGPW effect needs to be considered in most clinical MRI applications, and including FGPW effect allows the detecting of slower diffusion that is often encountered in polymer systems. Additionally, the spin-spin relaxation effect can be analyzed, which provides a broad view of the dynamic process in materials. The modified-Bloch equation based on the fractal derivative could provide a fundamental theoretical model for PFG anomalous diffusion. Introduction Over the last few decades, significant amounts of anomalous diffusion studies have been carried out in a variety of systems such as in polymer or biological systems.1,2,3,4,5,6 Anomalous diffusion can be detected by pulsed field gradient (PFG) diffusion experiments.
    [Show full text]