Physical fundamentals of magnetic resonance imaging

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1 Physical fundamentals of magnetic resonance imaging Stepan Sereda University of Bonn 1 / 26

2 Why? Figure 1 : Full body MRI scan (Source: [4]) 2 / 26

3 Overview Spin angular momentum Rotating frame and interaction with RF field Bloch equations & relaxation Detection of Signal RF pulses Spin Echo 3 / 26

4 MRI - Magnetic Resonance Imaging Sources of contrast (depend on tissue): proton density density of proton spins relaxation times T 1 and T 2 mobility of protons 4 / 26

5 Historical Overview (main moments) 1936: Gorter: First aasumption about resonance of nuclear susceptibility for RF fields 1938: Rabi s A new method of measuring the nuclear magnetic moment 1948: Percell and Ramsey put heads in an RF coil 1950: Hahn - Spin Echo 1966: Ernst and Anderson used pulses of RF 1971: Damadian showed difference in the T 1 relaxation times of cancerous and normal tissue 1973: Moon and Richards performed measurements of 31 P in intect blood 5 / 26

6 Spin angular momentum Need to align spins with the help of static magnetic field B 0 : Magnetic dipole moment: µ = γ I (1) Gyromagnetic ratio: γ = e 2m p (2) With B 0 : Equation of motion: d µ dt = γ µ B 0 (3) Larmor frequency: ω 0 = γ B 0 (4) 6 / 26

7 Magnetisation Magnetisation: M = Equation of motion: d M dt i µ i V (5) = γ M B (6) Figure 2 : Spin angular momentum I without (a) and with (b) magnetic field (Source: [1]) 7 / 26

8 Vector model Figure 3 : (a) Dipole magnetic moments at thermal equilibrium (b) Magnetisation vector M 0 (Source: [1]) 8 / 26

9 Basic Quantum-Mechanical Description I = h I(I + 1) cosβ = m z h I (7) E = µ B 0 = γ hm z B 0 ω 0 = γ B 0 (8) Figure 4 : (a) Possible spin states for I = 1 (b) 3D case (Source: [1]) 9 / 26

10 RF field and rotating frame Need Radio Frequency pulce B 1 ( B 0 ) to tilt M into xy-plane. B 1 rotates in xy-plane with ω 1 Rotating Frame In rotating frame: If ω frame = ω 0 B eff = (B 0 ω frame ) γ k (9) B eff = 0 (10) After B 1 application: d M dt rotation of M in rotating frame: = M γ(b 0 k + B1 ) (11) where T - time of rotation, α - flip angle α = γb 1 T (12) 10 / 26

11 Rotating frame Figure 5 : Longitudinal magnetisation M in the presence of RF field in (a) Laboratory frame S (b) Rotating frame S (Source: [1]) 11 / 26

12 Rotating frame Figure 6 : Precession of M around Beff ( ω γ B 0 ) (Source: [1]) 12 / 26

13 Bloch Equations & Relaxation After application of B 1 Bloch equations yield: dm z dt = (Mz M 0) T 1 dm x dt = Mx T 2 = My dm y dt T 2 (13) where T 1 - longitudinal and T 2 - transverse relaxation times solution: {M z (t) = M 0 + (M z (o) M 0 )e t T 1 M xy (t) = M xy (0)e t (14) T 2 for example measure Free Induction Decay (FID): where ρ - spin density F ID ρe t T 2 (15) 13 / 26

14 Longitudinal and Transverse Relaxation Longitudinal Relaxation T 1 - spin-lattice relaxation, exchange of energy between spin system and lattice ( 1s) Transverse Relaxation T 2 - spin-spin relaxation, loss of magnetisation in transverse plane ( 2 100ms) Dephasing Non same precession frequencies Non uniform B0 1 T = 1 2 T T 2 Information out of T 1 and T 2 Water bounding and absorption in a tissue Susceptibility and field inhomoginity Temperature, concentration, viscocity 14 / 26

15 Transverse relaxation At P - time varying magnetic fields because of µ i Figure 7 : Dipole interaction (Source: [1]) 15 / 26

16 Statistical distribution of spin states n n = e E kts (16) almost all spins in spin-up state very big decay time p = µ 0ω 3 0 γ2 h 6πc 3 (17) Figure 8 : (a) Energy levels for I = 1/2 (b) Difference in population between spin-up and spin-down (Source: [1]) 16 / 26

17 Detection of Signal Figure 9 : (a) Solenoid coil for creation of linear polarized RF field (b) Linearly polarized field decomposed into two counter-rotating circular components (Source: [1]) 17 / 26

18 RF pulse 1) Saturation-recovery (SR): Θ = 90 where T R -repetition time M z (t) = M 0 [1 e t T 1 ] (18) M z (0 ) = M 0 [1 e T R T1 ] (19) 2) Inversion-recovery (IR): Θ = 180 Θ = 90 where T I -inversion time M z (T I ) = M 0 [1 2e T I T 1 ] (20) 18 / 26

19 RF pulse T 1 measurement T 1 weighted images (T R < T 1 ) different T R and fitting Figure 10 : Basic sequence (a) SR (b) IR (Source: [1]) 19 / 26

20 Spin Echo T 2 measurement T 2 weighted images (T E - echo time, short) Need spin echo to get rid of T 2 different T E and fitting Figure 11 : Basic SE pulse sequence (Source: [1]) 20 / 26

21 Spin Echo Figure 12 : Detailed explanation of SE formation (Source: [1]) 21 / 26

22 Measurement of T 2 and Diffusion M y (nt E ) = ( 1) n M y (0)e nt E T 2 γ 2 G 2 DnT 3 E (21) where D - self-duffusion coefficient (describes diffusion of spins) different T E fitting Figure 13 : Carr-Purcell SE train (Source: [1]) 22 / 26

23 T 1 and T 2 weighting Figure 14 : T 1 and T 2 weighting (Source: [3]) 23 / 26

24 Summary Spins and static magnetic field B 0 RF pulse B 1 and Rotating frame Relaxation times T 1 and T 2 Basic pulses for T 1 measurements Spin Echo for T 2 measurements 24 / 26

25 Thank You for your Attention! 25 / 26

26 References [1] Webb s physics of medical imaging, 2nd edition, edited by M. A. Flower 2012, bpp [2] Paul S. Tofts Methods for quantitative relaxation parameter mapping: measuring T 1 and T 2, ISMRM 2009 [3] EECS University of Michigan, Lecture notes Medical Imaging Systems ( - visited ) [4] /11/Wellness-MRI-3_0T.jpg - visited / 26