qMRI mOOC · Chapter 3

T2 Mapping

Mono-exponential · Multi-exponential · T2*

Slides from the qMRLab mOOC chapter by Samuelle St-Onge
NeuroPoly Lab, Polytechnique Montréal

What is T2?

  • T2, the transverse or spin–spin relaxation time: spins dephase, so the magnetization in the x–y plane decays
  • The upper limit of the signal decay time under ideal imaging conditions
  • In practice, magnetic field inhomogeneities make the transverse magnetization decay faster (Chavhan 2009; Cohen-Adad 2014)
    • Macroscopic: metallic implants, air–tissue interfaces
    • Microscopic: differences in magnetic susceptibility between tissues
  • Transverse relaxation including these effects is T2*

Why map T2?

  • A T2 map is an image of T2 relaxation times, with insights into tissue composition
  • Sensitive to microstructural changes associated with disease: iron accumulation, myelination, inflammation (Dortch 2020)
  • A promising modality for clinical and research applications
  • Newer techniques go beyond the conventional ones developed for NMR, e.g. MR fingerprinting: several parameters from a single acquisition (Ma 2013; Hamilton 2017)

What this chapter covers

Mono-exponential

The voxel's signal arises from a single tissue compartment.

One T2 per voxel, fitted from a multi-echo spin echo acquisition.

Multi-exponential

Several tissues contribute to one voxel's signal.

Applications such as myelin water fraction (MWF) imaging.

T2*

Fundamentals of T2* mapping, and how its decay curve differs from T2.

Related application: quantitative susceptibility mapping (QSM).

T2 mapping vs T2-weighted imaging

T2-weighted (T2w)

  • The standard practice in clinical MRI
  • Tissues give signal according to their T2: differences in T2 provide contrast
  • Qualitative: interpretation depends on the observer

T2 mapping

  • Quantitative: fits multiple data points to a decay curve to calculate T2
  • Objective: disease progression monitoring, novel biomarkers
  • Consistent comparisons across imaging protocols: suited to longitudinal and multi-institutional studies (Cheng 2012)

T2-weighted images vs the T2 map

Use the menu to switch echo time.

Method 1 · Mono-exponential

Mono-exponential T2 Mapping

A single fitted T2 value for each voxel

Acquiring a decay curve

  • The simplest technique: several images with a single-echo sequence at different echo times (TE), then fit T2 to the decaying signal (Milford 2015)
    • Not typically used in vivo: prohibitively long acquisition times
  • Multi-spin echo (MSE) sequences, based on the Carr–Purcell–Meiboom–Gill (CPMG) pulse train (Carr 1954; Meiboom 1958)
    • Multiple 180° refocusing pulses generate multiple echoes in a single acquisition (Fatemi 2020; Milford 2015)
  • Substantially shorter scan time than separate single-echo acquisitions: MSE is the sequence of choice in clinical settings

Carr–Purcell, and the Meiboom–Gill improvement

Carr–Purcell

  • A 90° pulse, then a series of 180° pulses along the same axis (e.g. x)
  • Mitigates diffusion effects by preventing phase accumulation
  • But imperfections in the 180° pulses accumulate: Mxy is lost faster than expected, and T2 is inaccurate (Brown 2014)

Meiboom–Gill

  • The 90° pulse along one axis (e.g. x), the 180° pulses along a perpendicular axis (e.g. y)
  • This phase cycling distributes the errors so they average out over time
  • A more stable and accurate echo train (Brown 2014)

Pulse sequence

Multi-spin echo sequence: a 90 degree excitation followed by a train of 180 degree refocusing pulses, each followed by an echo, with the echo amplitudes decaying with T2.

Figure 3.2. Simplified illustration of a multi-spin echo sequence for T2 mapping based on the Carr–Purcell–Meiboom–Gill method.

The mono-exponential assumption

  • Homogeneity within each voxel is assumed implicitly: one T2 decay time across the tissue
  • So a single fitted T2 value for each voxel in the image
  • Widely used, but shown to be insufficient for proper estimation of tissue T2: a voxel's signal often arises from multiple components with different T2 values (Graham 1996)

Multi-exponential methods, and their use for myelin water fraction imaging, come later in the chapter.

Signal modelling

The decay of the transverse magnetization is exponential, from the transverse component of the Bloch equations:

$$M_{xy}(TE) = M_z(0^{-})\,e^{-TE/T_{2}}\qquad\text{(3.1)}$$
  • Mz(0−) is the longitudinal magnetization immediately before the 90° excitation pulse
  • Assumes the measured signal is proportional to Mxy, and that Mz(0−) is constant regardless of TE (Dortch 2020)

T2 and T2* decay in white and gray matter

Use the menu to show T2, T2* or both.

T2 and T1

  • In NMR physics, T2 must be equal to or shorter than 2 T1 (Levitt 2008)
  • T2 can exceed T1 in very rare cases (Traficante 1991)
  • In living organisms, T2 is always shorter than T1

Data fitting

The mono-exponential T2 signal decay:

$$S(TE) = S_{0}\,e^{-TE/T_{2}}\qquad\text{(3.2)}$$

S0 is the signal intensity immediately following the excitation pulse (Dortch 2020; Milford 2015).

Stimulated echoes

  • B1 inhomogeneities and RF pulse imperfections make refocusing pulses deviate from the ideal 180°
  • This generates additional echoes, known as stimulated or spurious echoes
  • They contaminate the signal decay: erroneous T2 estimates (McPhee 2018)

Fitting with improved accuracy

  • Use only the even-numbered echoes (Focke 2011; Kim 2009)
  • Or discard the first echo (Biasiolli 2013; Milford 2015)

T2*: faster than T2

  • So far, Mxy decays exponentially with TE / T2 (Eq. 3.1)
  • In practice, B0 inhomogeneities cause a faster loss of the transverse signal: T2* relaxation (Figure 3.3)
$$\frac{1}{T_{2}^{*}} = \frac{1}{T_{2}} + \frac{1}{T_{2}'}\qquad\text{(3.3)}$$
  • T2′ quantifies the part of the relaxation that is due to magnetic field inhomogeneities (Brown 2014)

T2′ and measuring T2*

  • T2′ mapping, by removing the T2 effect from T2*, could help brain disease diagnosis:
    • quantifying blood oxygenation (Lee 2014)
    • predicting infarct growth in acute stroke (Siemonsen 2008)
  • T2* is fitted with the same methods as T2:
$$S(TE) = S_{0}\,e^{-TE/T_{2}^{*}}\qquad\text{(3.4)}$$
  • Acquired with gradient echo (GRE) sequences, which are sensitive to field inhomogeneities, unlike the spin echo sequences used for T2 (Cohen-Adad 2014; Markl 2012)
  • SAGE (spin and gradient echo) acquires T2 and T2* simultaneously (Stokes 2014)

Noise

  • Noise makes the T2 decay harder to fit accurately, a problem when clinical contexts need highly precise T2 values
  • Particularly for pixel-wise T2 mapping: SNR is much lower than with region-of-interest (ROI) approaches (Sandino 2015)

Next: a single-voxel fit, true T2 = 109 ms, at four SNR levels.

Impact of noise on the T2 fit

Use the menu to change the SNR. The fitted T2 is in the legend.

How many echoes, and when?

  • The number of echoes depends on echo spacing, the risk of heating the sample, and processing low-SNR data (Shrager 1998)
  • The Cramér–Rao lower bound (CRLB): the smallest possible variance of an unbiased estimator, given the noise in the data (Cavassila 2001)
    • Gives the optimal number of echoes to fit the decay accurately (Jones 1996)
  • Optimized echo times for a predetermined range of expected T2 values beat uniformly spaced echo times in accuracy (Shrager 1998)

Benefits and pitfalls

Benefits

  • Simple, straightforward implementation: a convenient and efficient T2 fit
  • Multi-echo spin echo (MESE) sequences significantly reduce acquisition time (Fatemi 2020; Milford 2015)

Pitfalls

  • One T2 per voxel over-simplifies the microstructure: a problem for complex tissues with several T2 components in one voxel
  • MESE is sensitive to RF pulse imperfections: B1 inhomogeneities and reduced flip angles overestimate T2 (Fatemi 2020)

Method 2 · Multi-exponential

Multi-exponential T2 Mapping

The T2 contribution of each tissue compartment within a voxel

One voxel, several compartments

  • The mono-exponential curve is a single-compartment tissue model: every component in a voxel has the same T2
  • But a voxel may contain different tissue compartments with different T2 times, so this assumption can give inaccurate fits
  • Example: a voxel at a tissue boundary in the brain holding both CSF and gray matter, the partial volume effect
  • Multi-exponential T2 mapping compartmentalizes the tissues inside a voxel, with applications such as myelin water fraction imaging

Signal modelling: a sum of exponentials

The transverse magnetization acquired at different TEs, as a sum of exponential decays:

$$M_{xy}(TE) = \sum_{i=1}^{N} M_{z,i}(0^{-})\,e^{-TE/T_{2,i}}\qquad\text{(3.5)}$$
  • Each term is the contribution of the ith tissue component to the overall decay (Collewet 2022; Dortch 2020)

Next: a single voxel of myelin water (MW) and intra/extracellular water (IEW), with mono-exponential curves for each and the multi-exponential curve for both.

Mono- vs multi-exponential: myelin water and IEW

Drag the slider to change the MW and IEW fractions.

Data fitting

Eq. 3.5 in terms of the initial amplitude Si of each tissue component:

$$S(TE) = \sum_{i=1}^{N} S_{i}\,e^{-TE/T_{2,i}}\qquad\text{(3.6)}$$

Si: initial signal amplitude of the ith component; T2,i: its T2.

For myelin water (MW) and intra/extracellular water (IEW), as in Figure 3.5:

$$S(TE)_{MWI} = S_{MW}\,e^{-TE/T_{2,MW}} + S_{IEW}\,e^{-TE/T_{2,IEW}}\qquad\text{(3.7)}$$

Example: multi-exponential T2 mapping of the spinal cord

Use the menus to switch echo time (left) and map (right).

Application: myelin water fraction (MWF) imaging

  • Demonstrates the clinical relevance of multi-exponential T2 mapping (Kumar 2012; MacKay 1994)
  • Myelin quantification could help identify biomarkers for diseases such as multiple sclerosis (MS)
  • MWF does not directly measure myelin: it quantifies the water trapped between the myelin sheaths, the primary component of myelin (Alonso-Ortiz 2015)

Why not T2-weighted images?

  • Hyperintense lesions in MS lack specificity: indistinguishable from inflammation, edema, axonal loss (Alonso-Ortiz 2015; Lee 2021)
  • MWF could show early microstructural changes in myelin, for early diagnosis and monitoring

The myelin water fraction

The proportion of the MRI signal from myelin water, relative to the total water content:

$$MWF = \frac{S_{MW}\,e^{-TE/T_{2,MW}}}{S_{MW}\,e^{-TE/T_{2,MW}} + S_{IEW}\,e^{-TE/T_{2,IEW}}}\qquad\text{(3.8)}$$
  • Total water: myelin water (MW) + intra/extracellular water (IEW), also called axonal water (Lee 2021)
  • Myelin itself is hard to image directly: its T2 is very short
  • Myelin-associated water has a slightly longer T2, around 20 ms (Lee 2021): still short, but measurable with standard spin echo sequences

Application: quantitative susceptibility mapping (QSM)

  • QSM measures variations in tissue magnetic susceptibility (Ruetten 2019; Wang 2015)
  • Quantifies paramagnetic and diamagnetic substances such as iron, calcium, oxygen and myelin (Ruetten 2019): insights into tissue physiology and integrity
  • Altered iron levels are linked to multiple sclerosis (Stephenson 2014) and Parkinson’s disease (Chen 2019): candidate biomarkers for neurodegenerative disorders

QSM: the phase holds the susceptibility

  • The MRI signal is complex: magnitude and phase
  • T1, T2 and T2* contrasts use only the magnitude; the phase carries information about tissue magnetic susceptibility
  • T2* includes T2′, the dephasing from field inhomogeneities, caused mainly by susceptibility differences between tissues (Ruetten 2019)
  • QSM combines the magnitude and phase of T2* acquisitions to map susceptibility within tissues (Shmueli 2020)

Next: susceptibility and B0 in a brain, simulated at 7 T.

A B0 field map and its susceptibility distribution

Hover to read values.

Benefits and pitfalls

Benefits

  • More accurate T2 for complex, heterogeneous tissue: each voxel is multi-compartmental, each tissue with its own T2
  • Particularly suited to MWF imaging, to understand demyelinating diseases such as MS (Alonso-Ortiz 2015)

Pitfalls

  • More complex than mono-exponential: longer acquisition times (Kumar 2012)
  • Sensitive to noise (Dula 2009), which can make accurate fits challenging

Summary

Mono-exponential

Simple and efficient; fast with multi-echo spin echo.

One T2 per voxel over-simplifies tissue; B1 errors overestimate T2.

Multi-exponential

Comprehensive and accurate for heterogeneous or pathological tissue, e.g. MWF.

Longer acquisitions; sensitive to noise.

T2*

T2 plus field inhomogeneities (T2′); measured with gradient echo.

With the phase, the basis of QSM.

The choice of mono- or multi-exponential depends on the clinical or research application and the complexity of the tissues studied.

Exercises and discussion

  • The chapter ends with four exercises that use the interactive figures and the notebooks (on Binder or in MATLAB)
  • Questions and answers: github.com/qMRLab/mooc/discussions

Open the exercises in the book →