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Magnetic Resonance Imaging

-the basics-

Sonja Lambertz

What is MRI?

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medical imaging technique based on NMR

Introduction: the hydrogen atom

● 1 electron

● nucleus: 1 proton

remember: total nuclear spin

𝐼 = 𝐼 𝑖 + 𝐿𝑖𝑖

intrinsic spin: 𝐼𝑝 =1

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● total nuclear spin of hydrogen: ½

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Introduction: magnetic moment

● magnetic moment:

𝜇𝐼=𝛾𝐼 𝜇𝐼,𝑧 = ±1

2𝛾ℏ

𝛾: gyromagnetic ratio (material dependant)

γ∗ =γ

2π𝑀𝐻𝑧 𝑇

nucleus

Introduction: in a magnetic field

● Larmor frequency

𝜔0 = 𝛾𝐵0

𝜔0 ∝ MHz

● Zeeman-effect

𝐸 = −𝜇𝑧 ∙ 𝐵0

∆𝐸 = 𝛾ℏ𝐵0

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𝜇𝑧 = −1

2𝛾ℏ

𝜇𝑧 = +1

2𝛾ℏ

Introduction: magnetization

● excess of nuclei parallel to 𝐵0:

𝑁−

𝑁+ = 𝑒 Δ𝐸/𝑘𝑇

𝑁−

𝑁+ 𝐵0 = 1 T, 𝑇 = 37°C ∝ 6,6𝑝𝑝𝑚

• causes net magnetization:

𝑀 = 𝑑𝜇

𝑑𝑉 𝑀𝑧 =

𝑁−−𝑁+ 𝜇𝑧

𝑉

𝑀𝑧 𝐵0 = 1T, 𝑇 = 37°C = 3 ∙ 10−3 A m

𝑀𝑥,𝑦 = 0 8

nuclear magnetic resonance

● apply RF-field in x-y-direction 𝐵𝑇 ∝ 1mT:

𝐵𝑥 = 𝐵𝑇 cos𝜔𝑇𝑡

𝐵𝑦 = 𝐵𝑇 sin𝜔𝑇𝑡

𝐵𝑧 = 𝐵0

● resonance condition 𝜔𝑇 = 𝜔0:

− magnetic moment forced out of equilibrium postition

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nuclear magnetic resonance

applet:

http://www.drcmr.dk/BlochSimulator/

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nuclear magnetic resonance

● rotating frame moving with 𝜔0 around 𝐵0:

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No RF-Field With RF-Field

𝜔𝑇 = 𝛾𝐵𝑇

nuclear magnetic resonance

● length of pulse defines angle:

●

𝜋

2-pulse: flips 𝑀 into x-y-plane

𝜏𝜋2 ∝ 300μs 𝐵𝑇 ∝ 10−5T

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𝜔𝑇 =𝑑𝛼

𝑑𝑡

𝛼 = 𝛾𝐵𝑇𝜏

Free Induction Decay

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● transverse magnetization decays after RF-pulse

spin-lattice-relaxation

● longitudinal magnetization 𝑀𝑧 recovers

𝑀𝑍 𝑡 = 𝑀𝑧,𝑒𝑞 − 𝑀𝑧,𝑒𝑞 −𝑀𝑧 0 𝑒−𝑡/𝑇1

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spin-spin-relaxation

● transverse magnetization 𝑀𝑇 decays due to

random, temporary spin-spin-interaction:

slows down speeds up

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spin-spin-relaxation

● decay of transverse magnetization 𝑀𝑇

𝑀𝑇 𝑡 = 𝑀𝑇 0 𝑒−𝑡/𝑇2

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𝑇2∗-decay

● in reality: transverse magnetization 𝑀𝑇 decays

with time 𝑇2∗, faster than 𝑇2:

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𝑇2∗ =

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𝑇2+

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𝑇2𝑖

● additional 𝑇2𝑖 caused by inhomogenities.

𝑀𝑇 𝑡 = 𝑀𝑇 0 𝑒−𝑡/𝑻𝟐∗

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Spin Echo

● rephasing of the spins using a 180°-pulse

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TE=echo time

measuring 𝑇1

● use inversion recovery:

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𝑀𝑧 𝑡 = 𝑀𝑧 0 1 − 2𝑒−𝑡/𝑇1

measuring 𝑇1

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weighted MRIs

𝑻𝟏 𝑻𝟐 relative

proton

density

distilled water 3 s 3 s 1.0

cerebro spinal fluid 3 s 300 ms 1.0

gray matter 1.2 s 60-80 ms 0.98

white matter 800 ms 45 ms 0.80

fat 150 ms 35 ms 1.0

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weighted MRIs

𝑇1-weighted 𝑇2-weighted proton density

weighted

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conclusion

● static magnetic field aligns spins

● net magnetization

● transverse RF-field induces nutation of

magnetization, if 𝜔0 = 𝜔𝑇

● measurable magnetization in transverse plane

● material dependant relaxation times 𝑇1, 𝑇2 and

proton density give contrast

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sources

● “MRI Scanner” Image from "Colour & Form Study." Colourful Language. N.p., n.d. Web. 26 Apr.

2013. <http://eleanormaclure.wordpress.com/2012/08/07/colour-form-study/>.

● “Knie MR” Image from "Magnetresonanztomographie." Wikipedia. Wikimedia Foundation, 20

Apr. 2013. Web. 26 Apr. 2013. <http://de.wikipedia.org/wiki/Magnetresonanztomographie>.

● “Nuclear Spin” Image from "Nuclear Magnetic Resonance." IMAIOS. N.p., n.d. Web. 26 Apr.

2013. <http://www.imaios.com/en/e-Courses/e-MRI/NMR>.

● “Guy in MRI” Image from "Open Air MRI and Diagnostic Imaging." Center In Amarillo Texas.

N.p., n.d. Web. 26 Apr. 2013.

<http://sites.securemgr.com/folder14715/index.cfm?fuseaction=browse>.

● “Zeeman Splitting” Image from "ThomasPomorski.com." ThomasPomorski.com. N.p., n.d. Web.

26 Apr. 2013. <http://www.personal.psu.edu/tzp124/assign6.html>.

● “Flip angle”, “FID” Images from "BME/EECS 516 Home Page." BME/EECS 516 Home Page.

N.p., n.d. Web. 26 Apr. 2013. <http://web.eecs.umich.edu/~dnoll/BME516/>.

● “Spin Echo”, “T2measurement” Images from Nuclear Magnetic Resonance. RWTH Aachen,

Feb. 2011. Web. 26 Apr. 2013. <http://institut2a.physik.rwth-

aachen.de/de/lehre/praktikum/Anleitungen/FK14-NMR_20110207.pdf>.

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sources

● "DRCMR." Bloch Simulator for Education in MRI and NMR. N.p., n.d. Web. 28 Apr. 2013.

<http://www.drcmr.dk/bloch>.

● "BME/EECS 516 Home Page." BME/EECS 516 Home Page. N.p., n.d. Web. 28 Apr. 2013.

<http://web.eecs.umich.edu/~dnoll/BME516/>.

● Lehnertz, Klaus Prof. Dr. "Magnetresonanztomographie/Magnetic Resonance Imaging." Uni

Bonn. 2013. Lecture.

● Dossel, Olaf. Bildgebende Verfahren in der Medizin: Von der Technik zur medizinischen

Anwendung. Berlin: Springer, 2000. Print.

● Demtroder, Wolfgang. Atome, Molekule Und Festko rper: Berlin: Springer, 2005. Print.

● Demtroder, Wolfgang. Kern-, Teilchen- Und Astrophysik: Berlin: Springer, 2010. Print.

● "Nuclear Magnetic Resonance." IMAIOS. N.p., n.d. Web. 28 Apr. 2013.

<http://www.imaios.com/en/e-Courses/e-MRI/NMR>.

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Bloch equations

𝑑𝑀𝑥′

𝑑𝑡= −

1

𝑇2∗𝑀𝑥

′ + 𝜔0 −𝜔𝑇 𝑀𝑦′ − 𝜔𝐹 sin𝜓𝑀𝑧

′

𝑑𝑀𝑦′

𝑑𝑡= − 𝜔0 −𝜔𝑇 𝑀𝑥

′ −1

𝑇2∗𝑀𝑦

′ +𝜔𝐹 sin𝜓𝑀𝑧′

𝑑𝑀𝑧′

𝑑𝑡= +𝜔𝐹 sin𝜓𝑀𝑥

′ −𝜔𝐹 cos𝜓𝑀𝑦′ −

1

𝑇1𝑀𝑧

′ −𝑀𝑧0′

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