Proton Density and Water Content

Jack Allen,

The term Proton Density (PD) refers to the concentration of proton spins that contribute to the MR signal. A greater PD will produce a greater MR signal. Signal strength also depends on relaxation rates and the receive coil gain [Tofts2003]. Quantitative MRI methods can be used to estimate Proton Density (PD) and Water Content (WC) from the estimation of longitudinal magnetisation Mz at fully-relaxed thermal equilibrium. In this state, Mz is maximised and denoted as M0. This is performed through post-processing and calibration steps. Note that measurements for M0-mapping are made at thermal equilibrium, rather the magnetisation is perturbed and a signal model is used to fit parameters to the data and derive M0. The magnetisation components are also mentioned in other posts on this site.

Estimating M0

Firstly, the equilibrium magnetisation (M0) must be estimated. This can be acheived by various pulse sequence types, including single-parameter gradient echo or spin echo acquisitions. Some multiple-parameter sequences also produce M0 maps, such as Magnetic Resonance Fingerprinting (MRF). Estimated M0 maps are often simply normalised, with no receiver bias correction or water content estimation.

Spin echo model for M0

As an example, here is the signal equation for a spin echo acquisition, where S is the signal and M0 is PD modulated by a scaling factor C. The model shows there is a dependency on the relaxation times T1 and T2, and the pulse sequence Repetition Time (TR) and Echo Time (TE):

\begin{equation} S=M_0 (1 -exp( \frac{-TR}{T_1} )) (exp( \frac{-TE}{T_2} )) \\ M_0 = C \cdot PD \end{equation}

To isolate the effect of PD on S, the effects of the other terms must be removed or at least minimised. The T1 and T2 weighting of S can be reduced by setting TR >> T1 and TE << T2. At this limit, S is approximately equivalent to M0.

\begin{equation} \begin{aligned} S &=M_0 (1 -exp( \frac{-TR \gg T_1}{T_1} )) (exp( \frac{-TE \ll T_2}{T_2} )) \\ &\approx M_0 (1 – 0) (1) \\ &\approx M_0 \end{aligned} \end{equation}

However, depending on the T1 and T2 of the sample it may be challenging to achieve negligible T1 and T2 contrast due to TE and TR restrictions imposed by the sequence design.

Multiple T2-weighted images may be acquired using a different TE for each image and a constant TR >> T1. The signal model in this case is effectively a T2 decay curve, which can be fitted to each voxel across the image series to estimate T2 and M0.

\begin{equation} \begin{aligned} S &=M_0 (1 -exp( \frac{-TR \gg T_1}{T_1} )) (exp( \frac{-TE}{T_2} )) \\ &=M_0 (1 -0) (exp( \frac{-TE}{T_2} )) \\ &=M_0 (exp( \frac{-TE}{T_2} )) \end{aligned} \end{equation}

The model derives S at the limit TE = 0.

\begin{equation} \begin{aligned} S &= M_0 (exp( \frac{-(TE=0)}{T_2} )) \\ &= M_0 (exp(0)) \\ &= M_0 \end{aligned} \end{equation}

Receiver coil sensitivity

The scaling factor C represents the smoothly-varying spatial modulation of the PD map by the profile of the receive coil gain B1– [Tofts2003, Cercignani2018]. At a field strength of 3T, B1– bias removal does not have a straightforward solution [Volz2012]. The bias may be left uncorrected if the image intensity is normalised and T1 and T2 measurements are the immediate focus of the imaging study. However, a method for correcting for the receiver bias at 3T has been evaluated in the literature [Abbas2015], which uses the ratio of images acquired using a head coil and a body coil as receivers. This method would likely increase the total scan time needed. It would be valuable for future research to explore an appropriate strategy for correcting receive bias in QMRI maps without significantly increasing the scan duration, either with the Abbas2015 method or by an alternative technique.

Calibration to pure water

To measure water content, PD measurements can be calibrated using a sample of pure water. In practice, brain PD measurements can be calibrated using voxels in the Cerebral Spinal Fluid (CSF) of the ventricles, an area with approximately the same proton concentration as pure water [Tofts2003].

Clinical applications

Calibrated Proton Density (PD) can be used to indicate the local concentration of water in the brain and track post-stroke tissue changes. Stroke damage causes the local water content to increase, meaning that even uncalibrated PD mapping can provide indirect measurements of oedema progression and the effect of medication [Tofts2003]. A PD-mapping technique with a measurement error smaller than the change in water content would allow the detection of oedema changes over a longitudinal study. Qualitative images are unsuitable for this, as the contrast would likely change between scans. Additionally, calibrated PD (water content measurements) can be directly compared with normal values for a given tissue type (e.g. grey matter).

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