Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Signal generation and gradient fields in MRI Maximilian Oehm Basics of imaging Relaxation and Measurement Fast pulse sequences Summary June 16, 2015 Signal generation and gradient fields in MRI Contents Maximilian Oehm Summary of physical fundamentals Motivation 1 Summary of physical fundamentals 2 Motivation Complex representation and gradient fields 3 Complex representation and gradient fields Basics of imaging 4 Basics of imaging Relaxation and Measurement 5 Relaxation and Measurement Fast pulse sequences Summary 6 Fast pulse sequences 7 Summary Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary of physical fundamentals ~ ∝ e~z • Magnetic field B (few T ) • Splits up energy levels + −N− • N N+ +N− ∼ 1ppm P ~ ~ = m~ ∝ B • M V • No measurement in z-direction possible Summary ~ • Precession around B ωL = −γBz Images from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals • RF-Pulse (MHz, 100m) with ω = ωL flips magnetization Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement • π/2-pulse → rotation by π/2 → xy -plane Fast pulse sequences Summary • π-pulse → rotation by π → negative z-direction • Rotating MT → signal in antenna Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Summary of physical fundamentals Maximilian Oehm Summary of physical fundamentals • Nuclei interact with environment (quantum theory) Motivation Complex representation and gradient fields • Spin-lattice interaction → longitudinal relaxation (T1 ∼ s) • Spin-spin interaction → transverse relaxation (T2 ∼ 10ms) Basics of imaging Relaxation and Measurement • T1 and T2 interesting material property Fast pulse sequences ~ inhomogeneous • Environment and B Summary → different ωL → dephasing Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals • Dephasing → faster decay (T2∗ ∼ 1ms) (FID) Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary 1 1 1 + = T2∗ T2 T2′ • Want to measure T2 → spin-echo-technique Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Summary of physical fundamentals Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Images from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Motivation Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary http://www.siemens.com/press/en/presspicture/?press=/en/presspicture/2012/healthcare/imagingtherapy-systems/him201211001/him201211001-05.htm&content[]=HIM&content[]=H&content[]=HC &content[]=HCIM Signal generation and gradient fields in MRI Motivation Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary http://en.wikipedia.org/wiki/Magnetic_resonance_imaging Signal generation and gradient fields in MRI Magnetization as a complex number Maximilian Oehm Summary of physical fundamentals • Measure MT as imaginary number Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement MT = Mx + My i • Rotation and phase → exponential function MT = MT′ exp[−(ω0 t + φ)i] Fast pulse sequences Summary • Mathematics easier • Measurement at one instant of time → |MT | and phase Signal generation and gradient fields in MRI Quadrature detector Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary UR/I = U1 · sin/cos(ω00 t) · U2 · sin((ω00 + ∆ω)t + φ)) ❤ ❤❤❤ ✭✭ 1 ❤❤❤ ✭✭✭ ✭❤ ✭ ❤ ✭ = U1 U2 [cos/sin(∆ωt + φ) ∓ cos /✭sin((2ω + ∆ω)t + ✭ 00 ❤ ✭ ❤❤φ)] 2 ❤ ✭✭ 1 S(t) = UR + UI i = U1 U2 exp((∆ωt + φ)i) 2 Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Gradient fields Maximilian Oehm Summary of physical fundamentals ~ • Add small B(x) = Gx x e~z Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary ω(x) = −γ(B0 + xGx ) ωD (x) = −γxGx Image from S. Webb: Webb’s physics of medical imaging Signal generation and gradient fields in MRI Simple imaging technique Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences By Gz y z Summary x No relaxation considered Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Selective excitation • Apply Gz during π/2 pulse • Idea: Excite slice, push rest out of resonance Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary • QT: T2 decay for MT A(ω) = T2 1 + (ω − ωL )2 T22 • Finite pulse → additional frequencies ωD • Solve Bloch equations for My = My 0 : MT′ (t) = i · γ · My′ 0 exp[iωD t]B̃T (ωD ) Signal generation and gradient fields in MRI Selective Select tive excitation Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Images from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Images from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Phase encoding Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Gy Basics of imaging y Relaxation and Measurement Gx Fast pulse sequences Summary S(t) = MT (kx , ky ) = Z Z Z Z z x MT (x, y ) exp(ω(x)t) exp(φ(y ))dydx MT (x, y ) exp(−ikx x) exp(−iky y )dydx Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Phase encoding • After π/2 pulse before measurement • Apply Gy for time Ty Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary • Phase difference φ = −γ · Gy · y · Ty S(t) = Z Z MT′ 0 (x, y ) · exp(−iγGy Ty y )dydx ky = γGy Ty • Control ky with Gy and Ty . To resolve ∆y : 1 γ = Gy ,max Ty ∆y π Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Frequency encoding • After phase encoding • Apply Gx while measuring S(t) • MT (x, y ) → ωD = −γGx x S(t) = Z Z MT′ 0 (x, y ) · exp(−iγGx xt) exp(−iky y )dydx kx = γGx t • Measure multiple kx Summary • Bandwidth of antenna for image size Lx : bandwidth = γGx Lx Signal generation and gradient fields in MRI Cartesian 2D imaging sequence ky Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields kx Basics of imaging Relaxation and Measurement 90° Fast pulse sequences Summary Gz Gy Gx measurement Signal generation and gradient fields in MRI Cartesian 2D imaging sequence ky Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields kx Basics of imaging Relaxation and Measurement 90° Fast pulse sequences Summary Gz Gy Gx measurement Signal generation and gradient fields in MRI Example for a measurement Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Surfing through k-space ky Maximilian Oehm Summary of physical fundamentals GxTx GyTy Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement 90° measurement Fast pulse sequences Summary Gz Gy Gx kx Signal generation and gradient fields in MRI Relaxation Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields • Till now did not consider relaxation • Disadvantages: • Signal decays Basics of imaging • Decay → damping Relaxation and Measurement • Signal → reduce error, measure k-space Fast pulse sequences Summary • Advantages: • Additional quantities → differentiate tissues Signal generation and gradient fields in MRI Measurement of T2 Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary MT = MT 0 (x, y ) · exp(−TE /T2 (x, y )) Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Measurement of T1 Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary MT = MT 0 (x, y ) · (1 − exp(−TR /T1 (x, y )) Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Measurement Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Proton density TR long TE short T1 -weighted TR short TE short T2 -weighted TR long TE long • Different parameters → different contrast of different tissues Summary • Strong proton-density-, T1 - and T2 -weight → all possible images Image from S. Webb: Webb’s physics of medical imaging Signal generation and gradient fields in MRI Turbo-spin-echo sequence Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI EPI sequence Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Image from O. Dössel: Bildgebende Verfahren in der Medizin Signal generation and gradient fields in MRI Summary Maximilian Oehm Summary of physical fundamentals • Measure complex signal Motivation • Imaging through sequence of gradient fields Complex representation and gradient fields • Selected excitation: push unwanted part out of resonance Basics of imaging • Phase and frequency encoding: generate natural FT Relaxation and Measurement • Inverse FT of signal → image Fast pulse sequences • Can measure proton-density-, T1 and T2 weighted images Summary • Fast MRI: Turbo-spin-echo and EPI sequence Signal generation and gradient fields in MRI Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary Thank You For Your Attention! Signal generation and gradient fields in MRI Literature Maximilian Oehm Summary of physical fundamentals Motivation Complex representation and gradient fields Basics of imaging Relaxation and Measurement Fast pulse sequences Summary DÖSSEL, O., Bildgebende Verfahren in der Medizin. Von der Technik zur medizinischen Anwendung, Berlin, Heidelberg: Springer, 2000 WEBB, S., Webb’s physics of medical imaging, 2nd edition, Boca Raton, London, New York: CRC Press, Taylor & Francis Group, 2012 MORNEBURG, H., Bildgebende Systeme für die medizinische Diagnostik, 3rd edition, München: Publicis MCD, 1995
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