Excitation is a necessary process for all MRI in order to create observable magnetization to image. The simplest RF pulses (hard pulses) excite all spins in an object, however, carefully designed RF pulses are necessary in order to select slices, invert or rephase (as in a spin-echo) spin within a slice, pre-saturate fat or specific spatial locations, etc. This presentation will describe the basic principles for the excitation process and methods for the design of RF pulses for use in MRI.

The Bloch Equations. The Bloch equations govern the behavior of the net magnetization in the presence of applied magnetic fields. The Bloch equations provide the “classical” description of motion of the magnetization vector and does not easily account for some quantum mechanical behavior seen in, for example, coupled spin systems. While simple concepts such as on-resonant energy absorption can help to explain the excitation of a plane of on-resonance spins in slice selection and more detailed analysis is necessary for describing the slice profile and multidimensional excitation. Thus, we start with the Bloch equations, which neglecting relaxation, is: $$\frac{d\mathbf{M}}{dt}=\mathbf{M} \times\gamma\mathbf{B},$$ where $$$\mathbf{M} =\begin{bmatrix}m_x \: m_y \: m_z \end{bmatrix}'$$$ is the magnetization and $$$\mathbf{B} =\begin{bmatrix}B_x \: B_y \: B_z \end{bmatrix}'$$$ is the applied magnetic fields. This vector equation, while complicated, merely dictates that the magnetization M will precess around any B field at frequency $$$\omega=\gamma|\mathbf{B}|$$$.

Rotating Frame of Reference. One of the more useful tools in simplifying MRI concepts is the rotating frame of reference. Here we consider that our coordinate system for observation of the magnetization is rotating at a frequency, $$$\omega_0=\gamma B_0 $$$. In particular, the coordinate system is rotating about the z-axis in the same direction that M rotates about B. The z coordinate does not change, but we now must define a new x and y coordinate system. The “laboratory” frame of reference is the usual frame of reference with coordinates (x, y, z). The “rotating” frame of reference has coordinates (x’, y’, z). If we have magnetization precessing at $$$\omega_0$$$, it will appear to be stationary in the rotating frame as shown in Figure 1. More precisely, the magnetization precessing in the transverse plane, $$$m_{xy}=m_x+im_y =m_0 exp(-i\omega_0 t) $$$, will appear stationary in the rotating frame: $$$m_{xy,rot}=m_{x,rot}+im_{y,rot} =m_0$$$. Conceptually, we can think of this as being similar to riding on a carousel. If we are on the carousel, other objects on the carousel appear stationary, but to someone on the ground, the objects are spinning by at $$$\omega_{carousel}$$$ ($$$\omega_0$$$). The z-component of the magnetization is the same in both frames of reference: $$$m_{z,rot}=m_{z}$$$.

In the rotating frame of reference, the magnetization is not precessing. Thus, the apparent or effective B in the rotating frame is $$$\mathbf{B}_{eff} = 0$$$. More generally, the rotating frame has an effective z magnetic field that is $$$\omega_{frame} / \gamma$$$ less than the applied magnetic field: $$$B_{z,eff}=B_z-\omega_{frame}/\gamma $$$ . Thus, assuming $$$\omega_{frame}=\omega_0$$$, we can relate the rotating and lab frame magnetization and applied fields using:

$$m_{xy}=m_{xy,rot}exp(-i\omega_0t),m_{z}=m_{z,rot}$$

$$B_{xy,eff}=B_{xy}exp(i\omega_0t),B_{z,eff}=B_{z}-\omega_{0}/\gamma$$

along with the Bloch equations in the rotating frame:

$$\frac{d\mathbf{M}_{rot}}{dt}=\mathbf{M}_{rot} \times\gamma\mathbf{B}_{eff}.$$

For signal reception, it is interesting to note that the receiver allows you to see magnetization in the rotating frame defined by the frequency of the receive demodulator and thus, we can define the MRI signal equation using $$$m_{xy,rot}(t)$$$.

Excitation. Recall, we said that the Bloch equation, which describes the motion of M in the presence of a B field, dictates that the magnetization will precess around the B field at frequency $$$\gamma\mathbf{B}$$$. Here, we have two applied field, $$$\mathbf{B}_0$$$ and $$$\mathbf{B}_1$$$, and determining the motion of M in this case can be quite tricky with the Bloch equations. But fortunately, we have a tool to make this analysis easier: the rotating frame. Let’s consider the magnetization starting in its equilibrium position aligned to $$$\mathbf{B}_0$$$ (z axis). RF excitation ($$$\mathbf{B}_1$$$) that is applied at $$$\omega_0$$$ in the lab frame will appear as a stationary $$$\mathbf{B}_{1,eff}$$$ is aligned to the x’ axis in the rotating frame. In the rotating frame, $$$\mathbf{B}_0$$$ disappears and one only has to consider $$$\mathbf{B}_{1,eff}$$$ and we see that $$$\mathbf{M}_{rot}$$$ will precess around x’ ($$$\mathbf{B}_{1,eff}$$$) at frequency $$$\omega_1=\gamma|\mathbf{B}_{1}|$$$, as shown in Figure 2.

Slice Selective Excitation. The most common excitation pulse used in MRI is the slice selective RF pulse, which is done by applying an frequency selective RF pulse in the presence of a slice selection gradient (commonly z-gradient). Here the resonance frequency varies in the z-direction and the bandpass RF pulse excites only those spins whose resonant frequency lies within the band. We will examine the Bloch equations for this specific case. We will let $$$B_1(t)$$$ be a time-varying, real-valued magnetic field rotating at $$$\omega_0$$$, then $$$\mathbf{B}_{1,eff}(t)=B_1(t)\mathbf{i'}$$$, where $$$\mathbf{i'}$$$ is the unit vector along the x' axis. A z-gradient is applied, so the component in the z-direction is $$$\mathbf{B}_z=(B_0+G_z\cdot z)\mathbf{k}$$$ and $$$\mathbf{B}_{z,eff}=(G_z\cdot z)\mathbf{k}$$$ where $$$\mathbf{k}$$$ is the unit vector along the z axis in both rotating and lab frames. The total applied magnetic field in the rotating frame is then:

$$\mathbf{B}_{eff}=B_1(t)\mathbf{i'}+(G_z\cdot z)\mathbf{k}.$$

The Bloch equation in the rotating frame for this case reduces to the following:

$$\frac{d\mathbf{M}_{rot}(t)}{dt}=\begin{bmatrix}0 & \gamma G_z z & 0 \\ -\gamma G_z z & 0 & \gamma B_1(t) \\0 & -\gamma B_1(t) & 0 \end{bmatrix} \mathbf{M}_{rot}(t).$$

What we would like to know is how the magnetization, $$$\mathbf{M}_{rot}(t)$$$, varies as a function of z position following the application of the specified $$$B_1 (t)$$$ field. This is, in general, a very difficult equation to solve because it is non-linear.

Small Tip Angle Approximation. One particularly useful approach to the solution to the above Bloch equation is to use the “small tip angle approximation.” Here, we assume that $$$B_1 (t)$$$ produces a small net rotation angle, say, 30 degrees or less. In this case, we can assume the z component of the magnetization, $$$m_z$$$, is approximately equal to $$$m_0$$$ during the entire RF pulse. Essentially, we are saying that $$$\cos(\alpha(t))\approx 1$$$ where $$$\alpha(t)$$$ is the instantaneous flip angle at any point in the RF pulse. If this is the case, then we can assume that $$$dm_z/dt =0$$$ and $$$m_z (t)=m_0$$$ during the RF pulse. The Bloch equation in the rotating frame is now:

$$\frac{d\mathbf{M}_{rot}(t)}{dt}=\begin{bmatrix}0 & \gamma G_z z & 0 \\ -\gamma G_z z & 0 & \gamma B_1(t) \\0 & \underline{0} & 0 \end{bmatrix} \begin{bmatrix}m_{x,rot} \\m_{y,rot} \\ \underline{m_0} \end{bmatrix}.$$

This nicely decouples $$$m_{xy,rot} $$$ from $$$m_z$$$ and allows us to write a simple differential equation in $$$m_{xy,rot}$$$. Note that we are now examining this expression as a function of the slice direction, z, that is $$$m_{xy,rot}(z,t) = m_{x,rot}(z,t) + i m_{y,rot}(z,t) = $$$. We then have:

$$\frac{dm_{xy,rot}}{dt}=-i\gamma G_z z m_{xy,rot} + i\gamma B_1(t)m_0 .$$

Observe that $$$\gamma G_z z$$$ is a constant with respect to time and thus we have a first order differential equation with a driving function $$$i\gamma B_1(t)m_0$$$. For initial condition, $$$m_{xy,rot}(z,t) = 0$$$, the solution to this differential equation at the end of the RF pulse (T) can be shown be:

$$m_{xy,rot}(z,T)=i \gamma m_0 e^{-i \gamma G_z zT} \int_{0}^{T} e^{i \gamma G_z z t} B_1(t)dt .$$

We now make a variable substitution $$$t'=t-T/2$$$. We can also assume that the RF pulse that is symmetrical (even) around T/2 and that it is zero outside of the interval [0,T]. The magnetization can now be described as:

$$m_{xy,rot}(z,T)=i \gamma m_0 e^{-i \gamma G_z z T/2} \int_{-\infty}^{\infty} B_1(t'+T/2) e^{i \gamma G_z z t'}dt'$$

which we we can observe is:

$$m_{xy,rot}(z,T)=i \gamma m_0 e^{-i \gamma G_z z\frac{T}{2}} \mathscr{F}^{-1} \{B_1(s+T/2)\}\mid_{\omega=\gamma G_z z}$$

which is the well know Fourier relationship between the RF pulse ($$$B_1(t)$$$) and the slice profile. The leading phase term ($$$-i \gamma G_z z\frac{T}{2}$$$) must be compensated for by using a rephaser gradient after the RF pulse. So, we now see that designing a slice selective pulse for small tip angle is a matter of defining an RF pulse that has a particular spectrum (Fourier transform) and recognizing that the spectrum and spatial profile are related via $$$\omega=\gamma G_z z$$$.

More complicated RF pulse designs, for example, for the so-called multiband RF pulses [Larson 2008, Moeller 2010], one can create RF pulses that are the sum of RF subpulses, each exciting a specific spectral band corresponding to a particular slice. The Fourier relationship is linear, which means that the sum of responses of the subpulses. This works when the small-tip angle approximate is valid - as a practical matter, most small tip RF pulses work well up to 90 degrees.

Large Tip Angle Pulse Design. The above approach for RF pulse design approaches is based on the small tip angle approximation, but the design of large tip-angle pulses is more complicated. There are many approaches for the design of 1D RF pulses for large tip angles, e.g. for slice selective inversions or spin-echo pulses. One of the more commonly used approaches is the so-call Shinnar-LeRoux (SLR) algorithm popularized by Pauly et al. [Pauly 1991] and there are many other approaches as well. Figure 3 shows how a Fourier-designed pulse (windowed sinc pulse) and a SLR pulse of equal length have similar shapes and similar slice profiles for a 90 degree RF pulse. This shows how small-tip pulses behave well at least up to 90 degrees. On the other hand, Figure 4 shows the results from an inversion pulse resulting from scaling a Fourier-designed pulse scaled up to 180 degrees and compares to an SLR-designed inversion pulse. Clearly, the small-tip assumptions that go into the Fourier approach are no longer valid for 180 degree pulses and other approaches are necessary.

Multidimensional Excitation and Excitation k-space. We can construct a more general case where the applied gradient fields are along multiple directions and possibly time-varying, that is, $$$G_z z$$$ is replaced with the net effect of all gradients $$$\mathbf{G}(t)\cdot \mathbf{x}$$$, and the applied RF fields $$$B_1(t)$$$ can be complex valued, as well. Here the small tip approximation again leads to a differential equation:

$$\frac{dm_{xy,rot}}{dt}=-i\gamma [ \mathbf{G}(t)\cdot \mathbf{x}] m_{xy,rot} + i\gamma B_1(t)m_0$$

which, for an initial condition with $$$m_0$$$ in the z-axis, has a solution:

$$m_{xy,rot}(\mathbf{x},T)=i \gamma m_0 \int_{0}^{T} B_1(t)e^{-i \gamma \left[ \mathbf{x} \cdot \int_{t}^{T} \mathbf{G}(s)ds\right]}dt.$$

In a manner similar to image k-space, we can define excitation k-space [Pauly 1998a] as:

$$\mathbf{k}(t)=-\gamma\int_{t}^{T} \mathbf{G}(s)ds,$$

and then

$$m_{xy,rot}(\mathbf{x},T)=i \gamma m_0 \int_{0}^{T} B_1(t)e^{i \gamma \left[ \mathbf{x} \cdot \mathbf{k}(t)\right]}dt .$$

Note that excitation k-space is a time reversed integral of the gradient waveforms, and this forms the pathway upon which the RF energy is deposited in k-space by $$$B_1(t)$$$. Note also that since k-space is a time-varying function, there is also in implicit weighting by the inverse of the velocity of through excitation k-space – a slow pathway deposits more B1 than a fast pathway. Thus, the resultant excitation pattern as two components: 1) a k-space sampling pattern and 2) the excitation weighting defined by the RF waveform ($$$B_1(t)$$$) as modified the k-space velocity. More specifically, the sampling pathway can be defined as $$$s(\mathbf{k})=\int_{0}^{T} \delta_3(\mathbf{k}(t)-\mathbf{k})\mid \dot{\mathbf{k}}(t)\mid dt$$$, where $$$\delta_3$$$ is an impulse function in 3 dimensions that defines the k-space path. The RF weighting term is defined as $$$W(\mathbf{k})=b_1(t)/\mid \gamma \mathbf{G}(t)\mid$$$, which is defined only on the k-space pathway. We can then recast the excitation pattern (shown in the previous equation in a time integral) in the k-domain as:

$$m_{xy,rot}(\mathbf{x},T)=i \gamma m_0 \int_{\mathbf{k}}^{} s(\mathbf{k}) W(\mathbf{k}) e^{i \gamma \left[ \mathbf{x} \cdot \mathbf{k} \right]}d\mathbf{k} .$$

Thus, for a fully sampled, non-crossing excitation k-trajectory (e.g. an Archimedean spiral) and a desired excitation pattern, $$$d(\mathbf{x})$$$ , one can define the RF waveform as:

$$B_1(t)=F\{d(\mathbf{x})\} \mid_{\mathbf{k}=\mathbf{k}(t)} \cdot \mid \gamma\mathbf{G}(t) \mid .$$

In Figure 5, we demonstrate the design of an RF waveform for a spiral excitation k-space trajectory with a Gaussian 2D target profile and the resultant small-tip excitation pattern.

Iterative Methods for Multidimensional RF Pulse Design. The above approach provides an analytical expression for multidimensional RF pulses for a particular k-trajectory and desired pattern, however, this approach has some limitations with respect to the kinds of trajectories that can be easily employed, accounting for magnetic field inhomogeneity, enforcing power constraints, or exploiting spatially limited objects. Here, we describe an iterative pulse design approach developed by Yip et al. [Yip 2005], which is based on a discretization of the above time domain integral:

$$m_{xy,rot}(\mathbf{x},T)\approx i \gamma m_0 \sum_j^{} B_1(t_j)e^{i \left[ \mathbf{x} \cdot \mathbf{k}_j\right]}\Delta t ,$$

which can be rewritten as a simple matrix equation:

$$\mathbf{m}=\mathbf{A}\mathbf{b} ,$$

where $$$\mathbf{m}$$$ is the resultant magnetization (stacked up into a vector), $$$\mathbf{b}$$$ is the RF pulse vector, and $$$\mathbf{A}$$$ is a system matrix with elements:

$$a_{ij}= i \gamma m_0 e^{i \left[ \mathbf{x}_i \cdot \mathbf{k}_j\right]}\Delta t ,$$

where $$$ \mathbf{x}_i $$$ are voxel locations in the object. It is also possible to include the magnetic field inhomogeneity into the $$$a_{ij}$$$ terms [Yip 2005].

Using this formalism, we can easily design an RF pulse as a statistical estimation (minimization) problem:

$$\hat{ \mathbf{b}}=\text{argmin}_\mathbf{b}\{ \parallel \mathbf{Ab-d} \parallel_W^2 + \beta R (\mathbf{b})\}$$

where $$$ \mathbf{d}$$$ is the desired spatial excitation pattern and $$$R( \cdot )$$$ is a regularization term that can penalize, for example, the sum of the squared RF pulse, which is related to power deposition (SAR). With this form, constraints can be incorporated to limit the peak or local RF power. The first term in the argument is a W-weighted 2-norm, where the weighting function can be used to ignore voxels outside of the body, e.g. “don’t care” regions. The formalism can handle arbitrary excitation k-trajectories, including variable density and crossing trajectories.

Large Tip Angle Extensions. Again, the multidimensional RF pulse design approaches are based on the small tip angle approximation, but the design of large tip-angle pulses is more complicated. For the design of large-tip angle, multidimensional pulses, Pauly also described an approach that can be used for a series of self-refocused k-trajectories [Pauly 1989b]. A variety of other approaches have been developed to handle the large-tip angle case for multidimensional excitation [Pruessmann 2000, Xu 2007, Xu 2008, Sesompop 2008, Grissom 2008, Grissom 2009], which include methods for incremental correction to small-tip designs and full non-linear design approaches. In the above sections, we described the Bloch equations in the rotating frame and how this simplifies analysis of excitation pulses. We then described a very powerful approximation – the small tip angle approximation, which allows the use of a Fourier interpretation. We described how larger tip-angle pulse are different and require a different set of tools. We then extended the small-tip angle approximation to multiple spatial dimensions, where the Fourier interpretation leads to the concept of excitation k-space and to direct and iterative RF pulse design approaches.