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Relaxometry Equations

This document describes the mathematical models used for T1 and T2 relaxometry in quantitative MRI.

T1 Relaxometry

Inversion Recovery

The inversion recovery (IR) sequence is the gold standard for T1 measurement. Following a 180° inversion pulse, the longitudinal magnetisation recovers according to:

\[ S(TI) = \left| M_0 \left( 1 - 2\alpha \exp\left(-\frac{TI}{T_1}\right) + \exp\left(-\frac{TR}{T_1}\right) \right) \right| \]

Where:

  • \(S(TI)\) is the measured signal at inversion time \(TI\)
  • \(M_0\) is the equilibrium magnetisation (proportional to proton density)
  • \(\alpha\) is the inversion efficiency (ideally 1.0 for perfect 180° pulse)
  • \(TI\) is the inversion time (ms)
  • \(TR\) is the repetition time (ms)
  • \(T_1\) is the longitudinal relaxation time (ms)

The absolute value accounts for magnitude imaging, which cannot distinguish positive and negative magnetisation.

Simplified Form (Long TR)

When \(TR \gg T_1\) (typically \(TR > 5T_1\)), the \(\exp(-TR/T_1)\) term becomes negligible:

\[ S(TI) \approx \left| M_0 \left( 1 - 2\alpha \exp\left(-\frac{TI}{T_1}\right) \right) \right| \]

Null Point

The inversion time at which the signal crosses zero (null point) is:

\[ TI_{\text{null}} = T_1 \ln\left(\frac{2\alpha}{1 + \exp(-TR/T_1)}\right) \]

For long TR: \(TI_{\text{null}} \approx T_1 \ln(2\alpha) \approx 0.693 \cdot T_1\) (when \(\alpha = 1\)).

Variable TR (VTR) Method

The variable TR method measures T1 by acquiring spoiled gradient echo images at multiple repetition times with fixed flip angle:

\[ S(TR) = M_0 \sin(\theta) \frac{1 - \exp(-TR/T_1)}{1 - \cos(\theta)\exp(-TR/T_1)} \]

Where:

  • \(\theta\) is the flip angle

For small flip angles or when \(TR \gg T_1\), this simplifies to:

\[ S(TR) \approx M_0 \left( 1 - \exp\left(-\frac{TR}{T_1}\right) \right) \]

Linearisation for VTR

The simplified VTR equation can be linearised as:

\[ \ln(M_0 - S(TR)) = \ln(M_0) - \frac{TR}{T_1} \]

Or rearranged as:

\[ \frac{S(TR)}{1 - \exp(-TR/T_1)} = M_0 \]

Variable Flip Angle (VFA) Method

For spoiled gradient echo acquisitions with fixed TR and variable flip angle:

\[ \frac{S(\theta)}{\sin(\theta)} = E_1 \frac{S(\theta)}{\tan(\theta)} + M_0(1 - E_1) \]

Where \(E_1 = \exp(-TR/T_1)\).

This linearisation allows T1 to be extracted from the slope:

\[ T_1 = -\frac{TR}{\ln(E_1)} \]

T2 Relaxometry

Mono-exponential T2 Decay

The transverse magnetisation decays exponentially following excitation:

\[ S(TE) = A \exp\left(-\frac{TE}{T_2}\right) + C \]

Where:

  • \(S(TE)\) is the measured signal at echo time \(TE\)
  • \(A\) is the signal amplitude at \(TE = 0\)
  • \(TE\) is the echo time (ms)
  • \(T_2\) is the transverse relaxation time (ms)
  • \(C\) is an optional baseline offset (noise floor)

Two-Parameter Model (No Offset)

When the baseline offset is negligible:

\[ S(TE) = S_0 \exp\left(-\frac{TE}{T_2}\right) \]

This can be linearised by taking the natural logarithm:

\[ \ln(S(TE)) = \ln(S_0) - \frac{TE}{T_2} \]

Multi-echo Spin Echo

For a Carr-Purcell-Meiboom-Gill (CPMG) sequence with \(n\) echoes:

\[ S(n \cdot \tau) = S_0 \exp\left(-\frac{n \cdot \tau}{T_2}\right) \]

Where \(\tau\) is the inter-echo spacing.

Stimulated Echo Contamination

In practice, imperfect refocusing pulses lead to stimulated echo contamination, which can be modelled as:

\[ S(TE) = A \exp\left(-\frac{TE}{T_2}\right) + B \exp\left(-\frac{TE}{T_1}\right) \]

T2* Decay

For gradient echo sequences, the observed decay includes contributions from field inhomogeneities:

\[ S(TE) = S_0 \exp\left(-\frac{TE}{T_2^*}\right) \]

Where:

\[ \frac{1}{T_2^*} = \frac{1}{T_2} + \frac{1}{T_2'} \]

And \(T_2'\) represents the reversible dephasing due to macroscopic field inhomogeneities.

Fitting Considerations

Weighted Least Squares

For both T1 and T2 fitting, weighted least squares can improve estimates by accounting for the signal-dependent noise:

\[ w_i = S_i^2 \]

Non-linear Fitting

When linearisation is not possible (e.g., IR with unknown \(\alpha\)), non-linear least squares minimises:

\[ \chi^2 = \sum_{i=1}^{n} \left( S_i - S_{\text{model}}(t_i; \boldsymbol{\theta}) \right)^2 \]

Where \(\boldsymbol{\theta}\) represents the model parameters.

Goodness of Fit

The coefficient of determination \(R^2\) and root mean square error (RMSE) quantify fit quality:

\[ R^2 = 1 - \frac{\sum_{i=1}^{n}(S_i - \hat{S}_i)^2}{\sum_{i=1}^{n}(S_i - \bar{S})^2} \]
\[ \text{RMSE} = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(S_i - \hat{S}_i)^2} \]

References

  1. Deoni SCL, Peters TM, Rutt BK. High-resolution T1 and T2 mapping of the brain in a clinically acceptable time with DESPOT1 and DESPOT2. Magnetic Resonance in Medicine. 2005;53(1):237-241.

  2. Barral JK, Gudmundson E, Stikov N, et al. A robust methodology for in vivo T1 mapping. Magnetic Resonance in Medicine. 2010;64(4):1057-1067.

  3. Poon CS, Henkelman RM. Practical T2 quantitation for clinical applications. Journal of Magnetic Resonance Imaging. 1992;2(5):541-553.

  4. Look DC, Locker DR. Time saving in measurement of NMR and EPR relaxation times. Review of Scientific Instruments. 1970;41(2):250-251.