This course will cover the basic elements of spatial localization, with special focus on how the physics of spin evolution connect to k-space. This perspective will then be used to describe various phenomena in MRI, including PSFs, Nyquist relations, and non-Fourier encodings.
This course will cover the physics and math of spatial encoding in MRI, as well as the deep connection these have to k-space. Topics include:
1) Brief review of the basic elements in an MRI sequence
2) Connections between MRI data and k-space
3) Point spread functions and Nyquist relations
4) Advanced topics in k-space, including discretization, regions of k-space (high, low, positive, negative etc), contrast mixing, and non-Fourier encodings.
Gradients
Linear gradients (aka MRI gradients) modulate the amplitude of Bz spatially and are described according to the direction in which Bz varies. Assuming a uniform spin species (i.e. water in the human body), a gradient causes the Larmor frequency to vary by location. If a narrow band of frequencies are excited while the gradient is applied, a slice of space is excited; if orthogonal gradients are then applied during readout, the collected signal provides spatial encoding in plane.
For in-plane encoding, the gradient creates frequency variation across the slice, so the signal emitted from each spin is a sinusoid with a frequency that reflects its position along that axis. The collected signal is a sum of all those sinusoids, but Fourier transform reveals what frequency components are hidden in the bulk signal, which reflects what spatial positions were occupied by spins in the object. Therefore, the Fourier transform of the data is a projection of the object along the direction of the gradient, and a series of projections can be used to generate a full image.
While this explains how a line of data connects to image space, it is of limited use for understanding non-projection imaging methods and many other phenomena of MRI. A more general way to understand how MRI data contributes to the image is to analyze the MRI signal point by point, rather than line by line. With this goal in mind, we next consider both the physical state of spins evolving under a gradient and why each MR datapoint samples a point in the k-space of the object.
A uniform density plane of spins being observed in the rotating frame appears static, but when an x-gradient is applied, the spins begin to develop a lag or lead in phase proportional to their x-position. If we imagine either turning off the gradient or freezing time when the range of phase across the FOV is n2π, the resulting Mx is sinusoidal across the field of view. Written in convenient spatial units, the resulting Mx follows a function like sin(nx), and the resulting My is cos(nx). In other words, Mx(x)+iMy(x) is identical to a complex kx=n basis function.
k-Space
The principles of Fourier transform promise that any object can be uniquely represented as a weighted sum of complex sinusoids. The result of a Fourier transform conveys how much of each sinusoid must be added to reconstruct the object. Fourier analysis also provides a formula to solve inverse problem, how much of a given sinusoid is contained in an object. The weighting needed on a given sinusoid is simply the dot product of that sinusoid with the object itself, i.e. multiplication of the object with an ideal unit magnitude sine function with some particular kx and ky, followed by integration over all space.
When spins have been modulated by gradients, we collect the bulk signal from an object whose spins have been wound to a given (kx,ky). The bulk magnetization measured when spins are in that state is mathematically identical to the dot product described above-- multiplication of the object with an ideal unit magnitude sine function with some particular kx and ky, followed by integration over all space. For this reason, the magnitude of the bulk magnetization acquired when the spins in an object are wound to a certain (kx,ky) sinusoid represents a sampling of how much of that (kx,ky) sinusoid is contained in the object, i.e., the datapoint represents the k-space of the object sampled at (kx ky). This perspective can be used to better understand non-Cartesian trajectories, PSFs, and Nyquist sampling.
Beyond Fourier Encoding
Linear gradient encoding is mathematically equivalent to a dot product with an ideal unit magnitude sine function, but modern MRI has begun to explore alternative encoding methods that do not instinctively lend themselves to interpretation in k-space. However, any arbitrary encoding can be represented as a sampling of k-space, albeit the sampling of a distribution in k-space, rather than a single point. This approach can be used to understand how coils can fill skipped lines in k-space, despite the fact that neighboring points in k-space are completely independent. More recently, this perspective has been used to develop nonlinear gradient trajectories that enhance the sampling of gaps in k-space for better accelerated imaging.
D. Nishimura, Principles of Magnetic Resonance Imaging, 2010.
Haacke, Brown, Thompson and Venkatesan, Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2014.