Nuclear Magnetic Resonance In Medicine

Rochelle Radzyminski
March 26, 2021

Submitted as coursework for PH241, Stanford University, Winter 2021


Fig. 1: This is a comparison of a computed tomography scan (a), a T1-weighted MRI scan, and a T2-weighted MRI scan of the brain. A hypointense lesion and hyperintense lesion are visible in (b) and (c), respectively. The brightness is inverted on the two scans due to different effects of relaxation times. (Source: Wikimedia Commons)

Nuclear physics is at the center of much of modern medicine. Magnetic resonance imaging (MRI) exploits the phenomena of nuclear magnetic resonance (NMR) to non-invasively image the human body with sub-millimeter resolution. [1] This diagnostic imaging technique therefore was originally called Nuclear Magnetic Resonance Imaging (NMRI), but was renamed MRI in the 1970s. The term "nuclear" was dropped due to the public's negative associations with the word. With the ability to provide unique disease-specific molecular information, however, this nuclear-based technology is widely used to understand molecular function and dysfunction, as well as probe organs, tissues and the skeletal system. [2]

In 1946, Felix Bloch and Edward Purcell discovered nuclear magnetic resonance. Though their experimental methods differed, Bloch and Purcell simultaneously observed the same phenomenon: the response of magnetic nuclei to a continuous radio frequency (RF) magnetic field. [3] NMR has since been widely studied in both theory and experiment, and now has pivotal applications to physics, chemistry, and medicine. NMR can be used to identify molecular structure and to understand the quantum mechanical properties of nuclei, such as spin. Its studies have collectively led to the revolutionary development of MRI. [4]

In 1950, Erwin Hahn pioneered a novel technique known as the pulse sequence method for observing magnetic resonance. The method is comprised of π and π/2 RF pulses that rotate the magnetization of the sample and enable observations of free induction decay (FID) and spin echoes, which consequently allow for measurements of two properties that are characteristic of the sample being studied. [5] The theory at the core of MRI is outlined below.


MRI takes advantage of the abundance of hydrogen atoms in the human body and essentially measures how much water is in a given tissue sample. Hydrogen nuclei are comprised of one proton, which is a fermion with spin 1/2. This means that it is a small quantum-mechanical bar magnet that may point its magnetic moment μ in only one of two directions: either parallel or antiparallel to an applied magnetic field B0. These two "quantum states" of the proton are usually denoted ↑ and ↓.

Practical NMR requires a fairly large magnetic field strength B0 to break the energetic degeneracy between the ↑ and ↓ states, a phenomenon known as the Zeeman effect. Most MRI scanners used in hospitals and medical research have a magnetic field strength of 1.5 Tesla or 3 Tesla. The ↑ state has its energy lowered by μB0, and the ↓ state has its energy raised by the same amount. The N0 protons of a given sample in thermal equilibrium obey the Boltzmann distribution, so the number of protons in the two energy states are [6]

N = N0 eμB0/kBT
eμB0/kBT + e-μB0/kBT
       N = N0 e-μB0/kBT
eμB0/kBT + e-μB0/kBT

The magnetization is then

M0 = (N - N) μ = N0μ tanh(μB0/kBT) ≅ N0μ2B0

If the magnetic field strength is now increased so as to make the magnetization Mz and then suddenly reduced again back to B0, the magnetization relaxes back to M0 according to

= M0 - M
          M(t) = M0 + (Mz - M0) e-t/T1

The parameter T1 characterizing this experiment is referred to as the spin-lattice relaxation time. It is one of two crucial numbers measured through NMR and is a property unique to different substances. [3] It is called the spin-lattice relaxation time for historical reasons having to do with the non-biological origins of NMR.

T1 is measured in practice not by this slow relaxation method, but rather by detecting the "linewidth" of RF absorption at the Larmor frequency

ωL = 2 μB

This is a quantum-mechanical process in which the radio-frequency magnetic field "flips" the proton from ↑ to ↓. Since this experiment detects only one type of atom - hydrogen - imaging a part of the body requires distinguishing its protons from those in the rest of the body. This is achieved by causing ωL to vary from place to place by means of a magnetic field gradient. As a result, protons in the brain will have a value of ωL different from those in the feet, or anywhere else. Nuclei will absorb RF signal with frequency equal to their own unique, resonant value of ωL.

The second key process that generates MRI images is called spin-spin relaxation. The associated quantity is the spin-spin relaxation time, denoted T2. This is the time it takes for the spins in a system to move out of phase with each other. Pulsed NMR measures T2 by using a transmitter coil to administer what is called a π/2 pulse: intense RF is applied at the resonant Larmor frequency ωL of the nuclei of interest, but only long enough to rotate them into the transverse plane. [3,6] Once rotated, they create a time-dependent magnetization described by the equations

= - Mx
+ ωL My
= - My
- ωL Mx

and given explicitly by

Mx(t) = M0 cos(ωLt) e-t/T2
My(t) = M0 sin(ωLt) e-t/T2

This generates a back-RF that can be measured. In contrast to the T1 case, however, the spin signal that disappears in a time T2 is not actually gone. One sees this by waiting some time t0 longer than T2 and then applying a π pulse, reversing all the spins. After another amount of time t0 has elapsed, the signal reappears, although this time attenuated by the factor exp(-2t0/T1). In this way, T1 and T2 can be distinguished.

Spin-spin relaxation time stems from magnetic dipole-dipole interactions between protons. In addition to B0, each proton also sees a small magnetic field produced by surrounding protons - each one itself a little magnet. The protons have random positions and thus random interaction strengths with each other. This leads to their precession frequencies deviating slightly from ωL. In a time T2, this causes them to get out of phase. The π pulse flips the system upside down, causing the spins that were ahead of the other spins to now be behind. They eventually catch up with the others and re-phase, causing the echo. [7]

Image Construction and Interpretation

MRI medical images are "maps" of T1 and T2 made through sections of the body. [8] The resulting images look like panels b and c in Fig. 1, which shows both T1-weighted and T2-weighted images of the brain. The significance of different relaxation times in the context of image formation is that tissues with shorter T1 values will be brighter in T1-weighted images.

Tissue T1 (ms) T2 (ms)
Grey matter 1150 100
White matter 800 80
Skeletal muscle 1000 35
Fat 250 60
Table 1: Typical relaxation times for different tissues subject to 1.5 T magnetic field. [12]

In T2-weighted images, the opposite is true; tissues with longer T2 will appear brighter. [9] In Fig. 1, the T1 and T2-weighted images have inverted brightness. Typical relaxation times for different tissues are presented in table 1. The relaxation times of grey matter are longer than those of white matter. The light grey region of the T1-weighted image is white matter while the dark regions are grey matter. In the T2-weighted image, white matter appears dark while grey matter appears bright. Through these processes, detailed images of the body are generated.


MRI maps the location of various molecules - determined by these two characteristic relaxation times - and generates images. One of the first medical uses of MR scans demonstrated that spin-lattice and spin-spin magnetic relaxation times differed between normal and malignant tissues. [4] With over 30,000 scanners in the world, MRI is undoubtedly one of the most powerful diagnostic tools in contemporary clinical medicine and a testament to the technological and scientific advancements that nuclear physics enables. [10,11]

© Rochelle Radzyminski. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


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[3] V. P. Grover et al., "Magnetic Resonance Imaging: Principles and Techniques: Lessons for Clinicians," J. Clin. Exp. Hepatol. 5, 246 (2015).

[4] J. D. Kaunitz, "Magnetic Resonance Imaging: The Nuclear Option," Digest. Dis. Sci. 63 1100 (2018).

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[6] J. P. Hornak, The Basics of MRI (Rochester Institute of Technology, 1996).

[7] D. McRobbie, E. A. Moore, and M. J. Graves, MRI: From Picture to Proton, 3rd Ed. (Cambridge University Press, 2017), pp. 124-143.

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[9] P. Bannas et al., "Primer on Magnetic Resonance Imaging of the Liver," Clin. Liver Dis. 4, 120 (2014).

[10] S. Sammet, "Magnetic Resonance Safety," Abdom. Radiol. 41, 444 (2016).

[11] E. J. R. van Beek et al., "Value of MRI in Medicine: More Than Just Another Test?" J. Magn. Reson. Imaging 49, No. 7, e14 (2019).

[12] I. Fernández-Barahona et al., "Iron Oxide Nanoparticles: An Alternative For Positive Contrast in Magnetic Resonance Imaging," Inorganics 8, 28 (2020).