Relaxation 11/26/2020 | Page 2

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Master‘s Program in Medical Physics 11/26/2020 | Page 1

Physics of Imaging Systems Basic Principles of Magnetic Resonance Imaging III

Prof. Dr. Lothar Schad

Chair in Computer Assisted Clinical Medicine Faculty of Medicine Mannheim University of Heidelberg Theodor-Kutzer-Ufer 1-3 D-68167 Mannheim, Germany [email protected] www.ma.uni-heidelberg.de/inst/cbtm/ckm/

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Relaxation 11/26/2020 | Page 2

Relaxation

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Magnetization: M and M 11/26/2020 | Page 3 z xy

longitudinal magnetization: Mz

transversal magnetization: Mxy

transversal magnetization: Mxy - phase synchronization after a 90°-pulse - the magnetic moments  of the probe start

to precede around B1 leading to a synchronization of spin packages → Mxy - after 90°-pulse Mxy = M0

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: M and M 11/26/2020 | Page 4 z xy

source: Schlegel and Mahr. “3D Conformal Radiation Therapy: A Multimedia Introduction to Methods and Techniques" 2007

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Longitudinal Relaxation Time: T1 11/26/2020 | Page 5 thermal equilibrium excited state

after 90°-pulse:

-N-1/2 = N+1/2 and Mz = 0, Mxy = M0 after RF switched off: - magnetization turns back to thermal equilibrium

- Mz = M0, Mxy = 0

→ T1 relaxation longitudinal relaxation time T1 spin-lattice-relaxation time T1

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Physical Model of T1 Relaxation 11/26/2020 | Page 6

J()  B0 - in a real spin system (tissue) every nuclei is surrounded by intra- and intermolecular magnetic moments - thermal motion (rotation, translation, oscillation)

leads to an additional fluctuating magnetic field Bloc(t) with typical spectral distribution J()

- longitudinal components of J() at 0 allow energy transfer 0 from the spin system to the “lattice”

→ T1 relaxation

- trajectory of the tip of magnetization vector in the laboratory system

source: Liang and Lauterbur. “Principles of Magnetic Resonance Imaging” 2000

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Phenomological Description of T1  B 11/26/2020 | Page 7 0

the longitudinal magnetization Mz relaxes exponential to the equilibrium state Mz = M0 with a typical time constant T1

dMz/dt = (M0 x B)z + (M0 – Mz)/T1 : Bloch equation with T1

with Mz = 0 at t = 0:

Mz(t) = M0 (1 – exp(-t/T1)) → solution of Bloch equation 1.0 (t) / / (t) z

0.63

0.5

-t/T1 typical T1-values in (1-e ) tissue:100 - 2000 ms 0 normalized signal: M signal: normalized M repetition time TR [s] 0T1 1 2 3

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: T1 Relaxation 11/26/2020 | Page 8

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Transversal Relaxation Time: T2 11/26/2020 | Page 9 after 90°-pulse: -N-1/2 = N+1/2 and Mz = 0, Mxy = M0 after RF switched off: - magnetization Mxy starts to rotate in the x,y-plane at Larmor frequency - all transversal components J() of the fluctuating magnetic field Bloc(t) result in a dephasing of Mxy → spin-spin interaction - mainly static frequency components J() of the fluctuating magnetic field Bloc(t) at  = 0 are contributing - no energy transfer in the spin system (entropy ) - no influence of T2 on T1, they are independent ! J()  B0 → T2 relaxation transversal relaxation time T2 spin-spin-relaxation time T2

J(=0) - although technical inhomogeneities of B0 cause dephasing of Mxy → T2* (effective relaxation)

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Physical Model of T2 Relaxation 11/26/2020 | Page 10

Mxy

thermal equilibrium

signal signal intensity time z z z z Mz

M = 0 Mxy Mxy xy y y y y x x x x

RF B0 90°- pulse

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: Spin Dephasing 11/26/2020 | Page 11

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Phenomological Description of T2  B 11/26/2020 | Page 12 0

the transversal magnetization Mxy relaxes exponential to Mxy = 0 with a typical time constant T2

dMxy/dt = (M0 x B)xy – Mxy/T2 : Bloch equation with T2

with Mxy = M0 at t = 0:

Mxy(t) = M0 exp(-t/T2)) → solution of Bloch equation

0 1.0 (t) /M (t) xy

-t/T2 e typical T2-values in 0.5 tissue: 50 - 100 ms 0.37 water: ~1000 ms normalized signal: M signal: normalized

T2 echo time TE [ms] 0 50 100 150 200

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Simultaneous T1 and T2 Relaxation 11/26/2020 | Page 13

T1- and T2*- relaxation are simultaneous processes T2* < T1

T1-recovery

time

T2*-decay

longitudinal: relaxation to thermal tothermal equilibrium relaxation longitudinal: transversal: dephasing of spin ensemble

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: T1 and T2 Relaxation 11/26/2020 | Page 14

source: Schlegel and Mahr. “3D Conformal Radiation Therapy: A Multimedia Introduction to Methods and Techniques" 2007

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad T1 and T2 Relaxation Times in-vivo 11/26/2020 | Page 15

tissue T2 [ms] T1 [s] at T1 [s] at T1 [s] at 0.5 T 1.0 T 1.5 T skeletal 47 ± 13 0,55 ± 0,10 0,73 ± 0,13 0,87 ± 0,16 muscle myocardium 57 ± 16 0,58 ± 0,09 0,75 ± 0,12 0,87 ± 0,14

liver 43 ± 14 0,33 ± 0,07 0,43 ± 0,09 0,50 ± 0,11

kidney 58 ± 24 0,50 ± 0,13 0,59 ± 0,16 0,65 ± 0,18

spleen 62 ± 27 0,54 ± 0,10 0,68 ± 0,13 0,78 ± 0,15

fata 84 ± 36 0,21 ± 0,06 0,24 ± 0,07 0,26 ± 0,07

grey matter 101 ± 13 0,66 ± 0,11 0,81 ± 0,14 0,92 ± 0,16

white matter 92 ± 22 0,54 ± 0,09 0,68 ± 0,12 0,79 ± 0,13

a more than one exponential component - T1 increases with B0 - T2 nearly independent of B0

Bottomley et al. Med Phys 1984

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad NMR History: Relaxation Times 11/26/2020 | Page 16

excitation refocusing pulse pulse Harvard Nicolaas Bloembergen

1948

Robert Pound

excitation refocusing pulse pulse

Edward Purcell

• characterized the relaxation times of the nuclear response signal in detail

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Comparison: CT and MRI 11/26/2020 | Page 17

CT CT WM: 1025 Hu GM: 1035 Hu }  = 1% CSF: 1000 Hu

r T2 T1 T2 T1 MRI WM: 90 ms 550 ms GM: 100 ms 1000 ms }  = 100% CSF: >1000 ms 2000 ms

patient: astrocytoma grade II - no bones + best soft tissue contrast + no radiation

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Bloch Equations with T1 and T2 11/26/2020 | Page 18

dMz/dt = (M0 x B)z + (M0 – Mz)/T1

dMxy/dt = (M0 x B)xy – Mxy/T2

rotating system:

laboratory system:

complex signal:

M = Mx + iMy

M = M0 exp(-iLt – t/T2)

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Complex Signal: Simulated FID 11/26/2020 | Page 19

exp(-t/T2) FID

complex FT

2 T2

absorption dispersion 2 L Mx: real part FT T2 My: imaginary part FT 2 Mx() = M0 T2  My() = M0 T2 ( - L) 2 2 L 2 2 1 + ( - L) T2 1 + ( - L) T2 light dispersion

source: Liang and Lauterbur. “Principles of Magnetic Resonance Imaging” 2000

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Macroscopic Effect: Diamagnetism 11/26/2020 | Page 20 - Lenz’s law: the induced current produces an own

magnetic moment  in a conductor opposite to B0

- most of biological tissues have diamagnetic properties

since the electron magnetization Me of the electron sheath is opposite to B0 due to Lenz’s law: B = 0(H + Me) Me =  H -6 with 0 = 1.257 10 Vs/A magnetic field constant -6 H2O = -0.72 10 magnetic susceptibility

- weaker B-field inside a diamagnetic sphere due to e--shielding which is very effective since

e- = 658 p

- intersection of different tissues creates additional

local field inhomogeneities of B0 can be “homogenized” by additional shim coils

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Microscopic Effect: Chemical Shift 11/26/2020 | Page 21

1H spectrum of ethanol - precession frequency of nuclei bound in a specific molecule is determined by the local magnetic field Bloc:

Bloc = B - B loc =  Bloc = (1 - )B

6 with  = 10 ( - ref)/0 the relative chemical shift [ppm]

-  ~ 10 ppm for 1H MRS  ~ 100 ppm for 13C, 19F, and 31P

- high resolution spectrum at B0 > 1.5 T with B/B0 < 0.1 - 0.5 ppm show multiplet splitting due to spin-spin coupling → domain of MRS

- in MRI only protons of water are imaged, chemical shift is not relevant ! exception: fat = 3.5 ppm (220 Hz) at 1.5 T

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Summary: FID and MRS 11/26/2020 | Page 22

simulated isochromates one isochromate simulated 31P absorption spectrum

PCr MRS

three isochromates ATP

many isochromates

source: Liang and Lauterbur. “Principles of Magnetic Resonance Imaging” 2000

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Summary: FID and MRI 11/26/2020 | Page 23

FID signal is the transient response of a spin system after RF excitation; FID is a complex signal with amplitude and phase

FID amplitude is dependent on many parameters like: flip angle, number of spins, and magnetic field strength

FID timing is dependent on the grade of local magnetic field inhomogeneities characterized by T2*:

1/T2* = 1/T2 + Bz with T2* < T2

T2*: the effective (local) T2 relaxation time T2 : the true T2 relaxation time

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Standard Techniques for T1 and T2 11/26/2020 | Page 24

Saturation-Recovery Sequence Inversion-Recovery Sequence Spin-Echo Sequence

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Saturation-Recovery Sequence 11/26/2020 | Page 25

• saturation-recovery sequence • 90°-pulse moves the longitudinal

magnetization M0 to the x-, y – plane → FID

• transversal magnetization Mxy decays with T2* • longitudinal magnetization starts to recover to

thermal equilibrium → Mz with T1

• after TR actual (reduced) magnetization Mz is moved to the x-, y – plane → FID • repeat measurement with different TR → T1 determination by S ~ r [1 - exp(-TR / T1)] with TR >> T2*

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Inversion-Recovery Sequence 11/26/2020 | Page 26

• inversion-recovery sequence • 180°-pulse invert the longitudinal

magnetization M0 to –M0 at the z-axes • longitudinal magnetization starts to recover to

thermal equilibrium → Mz with T1 • inversion time TI • after TI 90°-pulse moves the actual (reduced)

longitudinal magnetization Mz to the x-, y – plane → FID

• transversal magnetization Mxy decays with T2* • repeat measurement with different TI → T1 determination by S ~ r [1 – 2 exp(-TI / T1)] with TR > 5 T1

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad T1 Measurement: Inversion Recovery 11/26/2020 | Page 27 inversion recovery (Mz(0) = -M0):

Mz(t) = M0 (1 – 2 exp(-TI/T1))

with Mz = 0 at TI = TI0: 0.5 = exp(-TI0/T1)

→ T1 = -TI0 / ln(0.5) = TI0 / 0.7

TI0

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad How to get rid of “Scanner’s” Dephasing ? 11/26/2020 | Page 28

180° refocusing pulse

90° 180° AQ t

to slow

to fast

to slow

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: Spin-Echo I 11/26/2020 | Page 29

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Spin-Echo Schema 11/26/2020 | Page 30

a) RF impulse schema

b) timing of longitudinal magnetization Mz

c) induced measured signal: spin-echo SE rephasing part dephasing part of signal of signal source: Schlegel and Bille. “Medizinische Physik Bd. 2” 2002

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Movie: Spin-Echo II 11/26/2020 | Page 31

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Multi Spin-Echoes 11/26/2020 | Page 32

RF excitation

MT signal

T2* : „effective“ relaxation with T2* < T2 1/T2* = 1/T2 + 1/T2´ T2 : „true“ relaxation due to irreversible dephasing T2‘ : „scanner“ relaxation due to static and constant field inhomogeneities source: Dössel. “Bildgebende Verfahren in der Medizin” 2000

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad NMR History: Spin-Echo 11/26/2020 | Page 33 Erwin Hahn Illinois • discovered a “second” nuclear resonance signal, the spin echo 1949 • achieved T1 and T2 weighting

excitation refocusing pulse pulse The first observed spin echo by E. Hahn (1950)

TE/2 TE/2

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad T2 Measurement by Multi Spin-Echo 11/26/2020 | Page 34 180° 180° 180° multi spin-echo technique 90° 90° signal signal signal TR: repetition time TE: spin-echo time time t

TE1 TE2 TE3

SI~Mxy

SE-signal: T2 ~ e-t/T2 T2* -t/T2 SI ~ Mxy = Mxy e

signal signal intensity time t WM: T2  90 ms GM: T2  100 ms CSF: T2 > 500 ms

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad T2 Measurement: SpinSpin-Echo Echo 11/26/2020 | Page 35

spin-echo (Mxy(0) = M0):

Mxy(t) = M0 exp(-t/T2)

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Summary: Spin-Echo 11/26/2020 | Page 36 z z z after 90°-pulse RF after 180°-pulse 180°-pulse

My y y y x x x

- spin-echo signal is the consequence of refocusing of a large amount of dephased isochromates

- spin-echo signal has maximum amplitude where isochromates reach new phase coherence

- spin-echo signal is a “two-sided” signal with two mirror-inverted FID’s, both components of the spin-echo increase/decay with T2*, but the amplitude of the spin-echo is T2-weighted

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Multi-Exponential T2: Tumor Tissue 11/26/2020 | Page 37

T1 T2m = rm exp(-t/T2m) spin density: r

M0 = r

T2b = rf exp(-t/T2f) + rs exp(-t/T2s)

most tissues in MRI: - bi-exponential due to partial volume effect

Schad et al. JCAT 1989

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad Multi-Exponential T1, T2: Fatty Tissue 11/26/2020 | Page 38

T1 T2m

T2f T2s

fatty tissue : bi-exponential in T1 and T2 tumor tissue : bi-exponential in T2 due to partial volume effect with CSF

Schad et al. MRI 1989

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad MRI – Based Therapy Planning ? 11/26/2020 | Page 39

tissues in NMR: - multi-exponential with very short T2 relaxation components → invisible at conventional MRI - measurement of proton densities (e.g. for n-dosimetry, HI-therapy) is not possible ! Brix. Dissertation, Heidelberg 1988

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Computer Assisted Clinical Medicine Prof. Dr. Lothar Schad T1-, T2 – Based Tissue Segmentation 11/26/2020 | Page 40 tissue characterization tissue segmentation

Schad et al. ZMP 1992 Friedlinger et al. Comp Med Ima Graph 1995

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