Signals

The measured MRI signal is determined by the free relaxation of tissue, but also by the impact of magnetic pulses and gradients applied during an MRI acquisition to generate transverse magnetization and localize the signals.

This section discusses the effect of these pulses on the tissue magnetization and the measured signal for common pulse sequences. See the table with definitions for a summary of relevant terms and definitions.

Definitions and notations

List of sequence parameters

Short name	Full name	Definition	Units	Sequences
TS	Sampling time	Duration of the signal readout for a single time point. If TS=0, the signals are sampled by interpolation. If TS is a finite value, the signals are averaged over a time TS around the acquisition time. Defaults to None.	sec	SS, SR
S0	Signal scale factor	Amplitude of the signal model	a.u.	SS, SR
FA	Flip angle	Angle of magnetization after excitation	degrees	SS, SR
TR	Repetition time	Time between excitation pulses	sec	SS, SR
TC	Time to center	Time between the preparation pulse and the k-space center	sec	SR
TP	Preparation delay	Time between the preparation pulse and the first k-space line	sec	SR
B1corr	B1 correction factor	Factor to correct flip angles for B1-variations	dimensionless	SR, SS

Sequence parameters

sequence	parameters
SS	S0, FA, TR
SR	S0, FA, TR, TC, TP

Spoiled gradient-echo steady state

The most common pulse sequence for T1-weighted DC-MRI involves applying selective radiofrequency pulses with a low flip angle (FA) in rapid succession, typically with a repetition time (TR) in the range of a few msec. After each pulse the signal is collected and any residual transverse magnetization is actively destroyed before applying the next pulse.

The pulses are initially applied to the equilibrium magnetization \(M_{ze}\), but after a short number of pulses a new equilibrium arises where the regrowth after each pulse exactly equals the reduction in \(M_z\) by the pulse itself. This is called the steady-state, and in a steady-state sequence, it is in this regime that data are collected.

Fast water exchange

We first consider the steady-state longitudinal magnetization in a tissue with fast water exchange (section Fast water exchange). If the longitudinal magnetization is \(M_z\) before a pulse with flip angle \(\alpha\), it has a value \(M_z\cos\alpha\) afterwards. Until the next pulse is applied, it relaxes freely to equilibrium. In the steady-state, the end result of that free relaxation is again the initial state \(M_z\) before the pulse. Using the explicit solution in Eq. (5) with \(M_z(0)=M_z\cos\alpha\), where \(\alpha\) is the flip angle:

\[M_z = e^{-T_RK}M_z\cos\alpha + \left(1-e^{-T_RK}\right)K^{-1} J\]

The parameters \(K, J\) are a function of \(R_1\) and are therefore changing during contrast agent injection. However, since the period of free relaxation TR is much shorter than typical T1-values, we have assumed here that \(K, J\) are constant during this time.

Solving the equation for \(M_z\) produces the steady-state magnetization under a series of repeated pulses:

\[M_z = \frac{1-e^{-T_R\,K}}{1-\cos\alpha\; e^{-T_R\,K}} K^{-1} J\]

In this context it is safe to ignore flow effects so that \(J/K\approx M_{ze}\), which is constant, and \(K\approx R_1\) (see section Fast water exchange). The signal itself is proportional to the transverse magnetization \(M_z\sin\alpha\) after the pulse, with a proportionality constant that depends on factors such as coil sensitivity. In practice all constants are assembled into a single scaling factor \(S_\infty\) to produce the final signal model for the spoiled gradient echo in steady state:

\[S = S_\infty\sin\alpha\frac{1-e^{-T_R\,R_1}}{1-\cos\alpha\; e^{-T_R\,R_1}}\]

In dcmri this model is implemented in the function dcmri.signal_ss:

S = signal_ss(Sinf, R1, TR, FA)

Here R1 is either a scalar, or a 1D array if it is variable. The other variables are scalar.

Restricted water exchange

If water exchange is restricted, the signal derivation is similar except that now we must use the vector form of the magnetization (see section Restricted water exchange):

\[\mathbf{M} = e^{-T_R\,\mathbf{K}}\mathbf{M}\cos\alpha + \left(\mathbf{I}-e^{-T_R\,\mathbf{K}}\right) \mathbf{K}^{-1}\mathbf{J}\]

The solution is also similar, though we must take care not to commute the matrices:

\[\mathbf{M} = \left(\mathbf{I} - \cos\alpha\, e^{-T_R\,\mathbf{K}}\right)^{-1} \left(\mathbf{I}-e^{-T_R\,\mathbf{K}}\right) \mathbf{K}^{-1}\mathbf{J}\]

As before the signal is proportional to the total magnetization with now takes the form \(\mathbf{e}^T\mathbf{M}\) with \(\mathbf{e}^T=[1,1]\). Assuming the equilibrium magnetization is the same \(m_e\) in both compartments we can extract it by defining \(\mathbf{j}=\mathbf{J}/m_e\) and absorbing the constant \(m_e\) in the global scaling factor \(S_\infty\):

\[S = S_\infty\, \sin\alpha\, \mathbf{e}^T \left(\mathbf{I} - \cos\alpha\, e^{-T_R\,\mathbf{K}}\right)^{-1} \left(\mathbf{I}-e^{-T_R\,\mathbf{K}}\right) \mathbf{K}^{-1}\mathbf{j}\]

If we ignore the inflow effects then \(\mathbf{j}\) is determined by relaxation rates \(R_{1,k}\) and volume fractions \(v_{k}\) of both compartments, and \(\mathbf{K}\) additionally depends on the water permeabilities \(PS_{kl}\) between the compartments.

This signal model is available in dcmri through the same function dcmri.signal_ss. The calling sequence is the same as in fast water exchange, except that now the volume fractions of the compartments need to be provided, and the water PS values across the barriers between them:

S = signal_ss(Sinf, R1, TR, FA, v, PS)

In this case R1 is a 2-element array, or a 2xn array with n time points if R1 is variable, v is a 2-element array, and PS is a 2x2 array with zeros on the diagonal and water PS values on the off-diagonal. The same function also applies when the number of compartments is larger than 2.

As mentioned flow effects can usually be ignored in steady-state sequences, but it is possible to include them by adding the water outflow from each compartment on the diagonal of PS, and providing the influx of normalized magnetization j with the same dimensions as R1:

S = signal_ss(Sinf, R1, TR, FA, v, PS, j)

Spoiled gradient-echo outside the steady state

If the magnetization is not in steady-state, then we have a different value \(\mathbf{M}_n\) before each pulse n, and we are left with an iteration:

\[\mathbf{M}_{n+1} = \cos\alpha e^{-T_R\,\mathbf{K}_n}\mathbf{M}_n + \left(\mathbf{I}-e^{-T_R\,\mathbf{K}_n}\right) \mathbf{K}_n^{-1}\mathbf{J}\]

Since we are considering larger time scales than the short interval TR, we accounted here for the possibility that the matrix \(\mathbf{K}\) depends on n as well. However, it is common at this point to assume that any time dependence of \(\mathbf{K}\) can be ignored. In an imaging context that is often justifiable when time scales are sufficiently short and the signal is dominated by a single time point corresponding to the center of k-space.

With a constant \(\mathbf{K}\), the difference from one iteration to the next can be written in terms of the steady-state magnetization \(\mathbf{M}_{ss}\):

\[\mathbf{M}_{n+1} - \mathbf{M}_n = \left(\mathbf{I}-\cos\alpha e^{-T_R\,\mathbf{K}}\right) \left(\mathbf{M}_{ss}-\mathbf{M}_n\right)\]

In terms of the difference \(\mathbf{D}_n=\mathbf{M}_n-\mathbf{M}_{ss}\) this reduces to:

\[\mathbf{D}_{n+1} = \cos\alpha e^{-T_R\,\mathbf{K}}\mathbf{D}_n\]

This equation is readily solved for any n. Reverting back to the magnetization in the solution gives:

\[\mathbf{M}_n = \mathbf{M}_{ss} + \cos^n\alpha e^{-nT_R\,\mathbf{K}} \left(\mathbf{M}_0-\mathbf{M}_{ss}\right)\]

Extracting the equilibrium magnetization by defining \(\mathbf{N}=\mathbf{M}/m_{e}\), and absorbing it in the scale factor again we have the formula for the signal of a spoiled gradient-echo sequence:

\[S = S_\infty\sin\alpha\,\mathbf{e}^T\left[ \mathbf{N}_{ss} + \cos^n\alpha\, e^{-nT_R\,\mathbf{K}} \left(\mathbf{N}_0-\mathbf{N}_{ss}\right)\right]\]