Advertiser Disclosure

This article may contain affiliate links. Which means if you click on a link and make a purchase, we may earn a commission at no extra cost to you. We only recommend products or services we believe may benefit the members of our community, and any opinions expressed are solely those of the author. Readers are encouraged to conduct their own research and exercise due diligence before purchasing or using any mentioned product or service. We sincerely appreciate your support.

mri physics, mri physics equations, larmor frequency, bloch equations, snr, signal to noise ratio mri

MRI Physics Equations

By

This helpful MRI technologist resource provides a detailed explanation of every MRI physics equation. From nuclear physics to MRI signal generation, we have it all covered.

Make sure to check out the related MRI technologists resources, such as the MRI scan parameters tradeoff chart, listed in the additional resources section below.

Table of Contents

MRI Physics Equations

Magnetic moment of a nucleus

The magnetic moment μ of a nucleus quantifies its interaction with an external magnetic field.

\Huge \mu = \gamma \hbar I

Where:

  • γ: Gyromagnetic ratio, specific to the nucleus (e.g., γ≈2.675×108 rad/s/T hydrogen-1).
  • ℏ: Reduced Planck’s constant (Defined as ℏ= (ℏ/2π), approximately equal to 1.055 x 10-34 joule-seconds)
  • (I): Nuclear spin quantum number (e.g., I = 1/2 for hydrogen-1).

The magnetic moment determines how strongly a nucleus responds to the external magnetic field B0, enabling nuclear magnetic resonance. For hydrogen protons, this is the basis for MRI signal generation.

This is the most fundamental, describing the intrinsic magnetic property of a nucleus (e.g., hydrogen proton) that makes NMR possible. It defines how nuclei interact with an external magnetic field.

Larmor Frequency

The Larmor frequency (ω0) in radians per second or f0 in Hertz) describes the precession frequency of nuclear spins (e.g., hydrogen protons) in a magnetic field B0.

\Huge \omega_0 = \gamma B_0

Where:

  • γ: Gyromagnetic ratio, a constant specific to the nucleus (e.g., γ/2π≈42.58 MHz/T for hydrogen-1).
  • B0: Strength of the external magnetic field (in Tesla)

Determines the resonance frequency for RF pulses to excite spins, critical for signal generation in MRI.

Similarly, the angular precession frequency equation (Larmor frequency, ω0) describes the rate at which nuclear spins precess around the external magnetic field B0. It’s critical for resonance and RF pulse design. Represented in hertz.

f_0 = \frac{\omega_0}{2\pi} = \frac{\gamma B_0}{2\pi}

Builds on the magnetic moment by defining the precession frequency of spins in a magnetic field B0. It’s essential for resonance and RF excitation.

NMR Signal Amplitude (Equilibrium Magnetization)

M_0 = \frac{N \gamma^2 \hbar^2 B_0}{4 k_B T}

Quantifies the net magnetization of spins at thermal equilibrium, which is the source of the MRI signal. Depends on the magnetic moment and B0.

Bloch Equations

The Bloch equations describe the time evolution of the magnetization vector M=(Mx, My, Mz) in a magnetic field B.

\frac{d\vec{M}}{dt} = \gamma \vec{M} \times \vec{B} - \frac{M_x \hat{i} + M_y \hat{j}}{T_2} - \frac{M_z - M_0}{T_1} \hat{k}

Where:

  • M: Magnetization vector with transverse (Mx, My) and longitudinal (Mz) components.
  • B: Total magnetic field (including static B0, RF pulses, and gradients)
  • T1: Longitudinal relaxation time (spin-lattice relaxation).
  • T2: Transverse relaxation time (spin-spin relaxation).
  • M0: Equilibrium magnetization.

The Bloch Equations govern how nuclear spins respond to RF pulses and relax back to equilibrium, forming the basis for MRI signal dynamics.

These equations describe the dynamics of magnetization (precession, excitation, and relaxation) under magnetic fields and RF pulses. It integrates the magnetic moment and Larmor frequency to model spin behavior.

T1 Recovery (Longitudinal Relaxation)

Describes the recovery of longitudinal magnetization Mz after excitation by an RF pulse.

M_z(t) = M_0 \left(1 - e^{-t/T_1}\right)

Where:

  • M0: Equilibrium magnetization.
  • T1: Longitudinal relaxation time, tissue-specific (e.g., ~1–2 s for tissues at 1.5 T).
  • ( t ): Time after excitation.

Determines the rate at which spins realign with B0 , affecting T1-weighted imaging contrast.

Details the recovery of longitudinal magnetization after RF excitation, a key component of the Bloch equations. It’s fundamental to signal evolution before imaging specifics.

T2 Relaxation

Describes the decay of transverse magnetization Mxy due to spin-spin interactions.

M_{xy}(t) = M_{xy}(0) e^{-t/T_2}

Where:

  • Mxy: Transverse magnetization in the xy-plane.
  • T2: Transverse relaxation time, tissue-specific (e.g., ~50–100 ms for tissues at 1.5 T).

Governs signal loss in the transverse plane, critical for T2-weighted imaging and echo formation.

Similarly,

S(t) = S_0 e^{-t/T_2}

Describes the decay of transverse magnetization due to both spin-spin interactions (T2) and magnetic field inhomogeneities.

Where:

  • T*2: Effective transverse relaxation time (1/T*2 = 1/T2 + 1/Tinhom).

Relevant in gradient echo sequences, where field inhomogeneities (e.g., from tissue interfaces) cause faster signal decay than T2 alone.

Describes transverse magnetization decay due to spin-spin interactions, another key component of the Bloch equations. (Consolidates “Transverse Relaxation” and “T2 Decay.”) It follows T1 as it’s part of relaxation dynamics.

T2 Decay

\displaystyle f_0 = \frac{\omega_0}{2\pi} = \frac{\gamma B_0}{2\pi}

T2 Decay Including field inhomogeneities

M_{xy}(t) = M_{xy}(0) e^{-t/T_2^*}

Extends T2 relaxation by including signal loss from magnetic field inhomogeneities, relevant for practical MRI signal behavior in gradient echo sequences.

RF Pulse Flip Angle

\alpha = \gamma B_1 \tau

Similarly,

The flip angle θ is the angle by which the magnetization is rotated by an RF pulse.

\alpha = \gamma \int_0^\tau B_1(t) \, dt

Where:

  • B1: Amplitude of the RF magnetic field.
  • τ: Duration of the RF pulse.

Controls the amount of excitation (e.g., 90° or 180° pulses) in MRI sequences.

Governs how RF pulses manipulate magnetization to create measurable signals. It’s applied after understanding equilibrium and relaxation dynamics. (Consolidates “Flip Angle.”)

Ernst Angle (for Gradient Echo)

The Ernst angle θ maximizes signal in steady-state gradient echo sequences with short TR.

\cos \alpha_E = e^{-TR/T_1}

Where:

  • (TR): Repetition time.
  • T1: Longitudinal relaxation time.

Optimizes signal intensity in fast imaging sequences like FLASH or GRASS

A specific application of the flip angle for optimizing signal in steady-state gradient echo sequences, building on RF pulse effects and T1 relaxation.

MRI Signal Induced In Coil (Faraday’s Law)

V(t) = - N A \frac{dB(t)}{dt}

Describes how the precessing transverse magnetization (post-RF excitation and relaxation) induces a detectable voltage in the receiver coil, the first step in signal acquisition.

MRI Signal Equation

The MRI signal (S(t)) is the Fourier transform of the spin density ρ(r), modulated by relaxation and phase encoding.

\displaystyle S(t) = \int \rho(\mathbf{r})\, e^{-t/T_2(\mathbf{r})} e^{-i \gamma \int_0^t \mathbf{G}(\tau) \cdot \mathbf{r}\, d\tau} \, d\mathbf{r}

Where:

  • ρ(r): Proton density at position r.
  • T2(r): Transverse relaxation time at position r.
  • G(τ): Gradient magnetic field vector.
  • γ: Gyromagnetic ratio.
  • e: Euler’s Number (mathematical constant approximately equal to 2.71828).

Describes how the MRI signal is collected in k-space, enabling spatial encoding and image reconstruction.

Combines proton density, T2 relaxation, and spatial encoding (via gradients) to describe the raw MRI signal in k-space, building on the induced voltage.

Spin Echo Time (TE) and Repetition Time (TR)

\text{Signal} \propto \left(1 - e^{-TR/T_1}\right) e^{-TE/T_2}

Where:

  • (TR): Repetition time, interval between successive RF pulses
  • (TE): Echo time, time between RF pulse and signal measurement.

(TR) and (TE) are adjusted to create T1-weighted (short TR, short TE), T2-weighted (long TR, long TE), or proton density-weighted images.

Specifies how sequence timing parameters (TE, TR) modulate T1 and T2 effects to produce contrast in the MRI signal, applying the signal equation in practical sequences.

Gradient Field

The magnetic field B(r) at position r is the sum of the static field B0 and the gradient field.

B(z) = B_0 + G_z z

Where:

  • G: Gradient vector (e.g., Gx, Gy, Gz) in T/m.

Gradients create spatially varying magnetic fields for slice selection, phase encoding, and frequency encoding.

Introduces spatial variation in the magnetic field, necessary for encoding spatial information in the MRI signal, setting the stage for imaging.

Slice Selection

Determines the position (z) of a slice selected by an RF pulse with frequency ω0.

\Delta z = \frac{\Delta f}{\gamma G_z}

Where:

  • Δz: slice thickness (m)
  • Δω\Delta\omegaΔω: angular frequency bandwidth (rad/s)
  • Δf\Delta fΔf: frequency bandwidth (Hz)
  • γ\gammaγ: gyromagnetic ratio (rad·s⁻¹·T⁻¹), for protons γ/2π≈42.577 MHz/T
  • Gz​: gradient strength along z (T/m)
  • B0​: main magnetic field (T)
  • ω0​, f0: Larmor frequency (rad/s and Hz, respectively)

Allows selective excitation of a specific slice in the imaging volume.

Uses gradients to excite specific slices, a fundamental step in spatial localization that builds on the gradient field concept.

Gradient Encoding (k-space)

The k-space vector k defines the spatial frequency sampled during MRI acquisition.

k(t) = \frac{\gamma}{2\pi} \int_0^t G(\tau) \, d\tau

Where:

  • G(τ): Gradient magnetic field as a function of time.
  • γ: Gyromagnetic ratio.

Gradients encode spatial information by altering the phase and frequency of spins, enabling image reconstruction via inverse Fourier transform.

Describes how gradients encode spatial frequencies in k-space, following slice selection as part of the imaging process.

Fourier Reconstruction

The image ρ(r) is reconstructed by performing an inverse Fourier transform on the k-space signal S(k).

\rho(x) = \int_{-\infty}^{\infty} S(k) e^{i 2\pi kx} \, dk

Where:

  • k: Spatial frequency vector.

Converts raw MRI data (k-space) into a spatial image, the final output of MRI.

Converts k-space data (from gradient encoding) into a spatial image, the final step in image formation.

Diffusion-Weighted Imaging (DWI)

Describes signal attenuation due to water diffusion in tissue.

B = \mu_0 (H + M) = \mu_0 (1 + \chi) H

Where:

  • S0: Signal without diffusion weighting.
  • (D): Apparent diffusion coefficient (ADC).
  • (b): b-value, determined by gradient strength (G), duration δ, and time between gradients Δ..

Used in DWI to detect restricted diffusion (e.g., in stroke or tumors).

A specialized technique that modifies the MRI signal to measure tissue diffusion, building on the signal equation and gradient encoding.

Magnetic Susceptibility

Describes the local magnetic field change ΔB due to tissue magnetic susceptibility χ.

\displaystyle f_0 = \frac{\omega_0}{2\pi} = \frac{\gamma B_0}{2\pi}

Where:

  • χ: Magnetic susceptibility of the tissue.
  • B0: External magnetic field.

Affects T2* decay and causes artifacts in regions with susceptibility differences (e.g., near air-tissue interfaces).

Addresses local field variations affecting T2* and image artifacts, relevant after understanding signal and imaging processes.

MRI Signal to Noise Ratio (SNR)

\text{SNR} \propto \frac{V_{\text{signal}}}{\sigma_{\text{noise}}}

A practical metric for image quality, considered last as it depends on signal generation (from earlier equations) and imaging parameters like voxel size and acquisition time.

Key Takeaways

It’s Important to keep in mind these equations assume idealized conditions. Real MRI systems in clinical settings account for additional factors like noise, field inhomogeneities, and hardware limitations. Additionally, tissue-specific values of T1, T2, and ρ (rho) create image contrast by influencing signal behavior.

Understanding these parameters helps MRI technologists optimize image quality and tailor protocols to clinical needs. Check out our related resources section below for additional guides, tools, and resources curated for our MRI technologist community members.

Larry Lopez

Written by

Larry Lopez

Larry is a biomedical imaging specialist with more than 16 years of professional experience in MRI, CT, and PET system installation, calibration, quality assurance, and advanced troubleshooting. As the founder, digital creator, and lead author of MRIPETCTSOURCE, he produces educational content designed to elevate the skills of technologists, engineers, and imaging center operators. Larry also serves as the chief technical advisor and lead web developer for MedicalImagingSource.com, where he oversees the accuracy, technical depth, and clinical relevance of all published resources. His work integrates field expertise with clear, evidence-based explanations to support both professionals and patients. Connect with Larry on social media: LinkedIn | YouTube | X (Twitter) | Instagram | Pinterest | Facebook

Read more on Larry’s author page.

Latest Articles

The information provided by MRIPETCTSOURCE (“we,” “us,” or “our”) on https://www.medicalimagingsource.com (the “Site”) is for general informational purposes only. All information on the Site is provided in good faith, however we make no representation or warranty of any kind, express or implied, regarding the accuracy, adequacy, validity, reliability, availability, or completeness of any information on the Site. UNDER NO CIRCUMSTANCE SHALL WE HAVE ANY LIABILITY TO YOU FOR ANY LOSS OR DAMAGE OF ANY KIND INCURRED AS A RESULT OF THE USE OF THE SITE OR RELIANCE ON ANY INFORMATION PROVIDED ON THE SITE. YOUR USE OF THE SITE AND YOUR RELIANCE ON ANY INFORMATION ON THE SITE IS SOLELY AT YOUR OWN RISK.

Amazon and the Amazon logo are trademarks of Amazon.com, Inc. or its affiliates.

Related Posts

Scroll to Top