The effect of any pulse sequence on the magnetization in an object can be predicted very accurately using the Bloch equation. A general algebraic inversion of the Bloch equation is not possible and thus, the full set of object and system properties and parameters cannot be derived from measurement data directly. Using a few assumptions and neglecting possible deviations, the results of a given pulse sequence can be calculated and the spatial encoding can be inverted to reconstruct an image. But what if these assumptions are wrong?
The effect of any pulse sequence on the magnetization in an object can be predicted very accurately using the Bloch equation. This requires, however, not only knowledge of the object and its spatially distributed MR properties, such as relaxation times T1, T2, T2*, magnetic susceptibility or chemical shift, but also information about local magnet field strength (B0-shim), magnetic field gradient distribution and timing, and local RF field strength (transmit field). Many of these parameters are not known a priory but are the desired result of a measurement, i.e. the object properties. A general algebraic inversion of the Bloch equation is not possible and thus, the full set of these properties and parameters cannot be derived from measurement data directly. Using a few assumptions and neglecting deviations from perfect homogeneity (B0 and B1) or linearity in space and time (gradients, B1), the contrast of a given pulse sequence can be calculated and the spatial encoding can be inverted to reconstruct an image. For many MR sequences, the effects on signal and contrast generation can be separated from image encoding.
For many pulse sequences, the image contrast can be calculated more easily if some assumptions are made. For example, in a (spoiled) FLASH sequence the transversal magnetization is assumed to be zero at the end of TR, leading to the known T1-weighting in dependence of TR. Similarly, the transition into the steady state in many sequences with short TR is neglected when their contrast is predicted. Often, the contrast prediction is based on the relaxation times alone, neglecting effects of off-resonance, chemical shift or magnetization transfer. The last effect can have a relevant impact on the image contrast in multi-slice sequences where slices are considered to be independent.
Apart from these effects on image contrast, the image geometry and fidelity are impacted through deviations from the assumptions in the image encoding process. The k-space formalism is a powerful tool and allows image reconstruction using simple Fourier transformation. It is assumed that the signal phase is given through the frequency variation induced by linear encoding field gradients. Thus the k-space position can be calculated as the time integral of the gradient. Any deviations in the k-space encoding, caused by additional unknown field variation (B0-shim), non-linearities of the field gradients, timing errors or eddy currents lead to corresponding errors in the image reconstruction. These commonly manifest as geometric distortions.
An additional difficulty is the interaction between the above mentioned signal variation throughout a sequence (due to transition into a steady state or encoding in multiple echoes) and the encoding order in k-space. Mathematically all points in k-space are identical and the reconstruction is independent of the encoding order. However, due to the typical compact shape of imaging objects the energy is not distributed equally throughout k-space. Therefore, the signal variation throughout a sequence leads to a strong dependence of the reconstruction on the signal distribution in k-space.
With the increase in computational power reconstruction of images through inversion of the full signal generation and image encoding process becomes more popular. Correction of and important insights into the different aspects of the deviation from the ideal signal generation and encoding can be gained when single effects are considered although interactions between the effects exist. Deviations from linear image encoding gradients lead to distortions depending on the acquisition bandwidth. With known gradient fields (one time calibration), the distortion can be calculated and images can be corrected through interpolation to the correct coordinates. Additional distortions due to B0 inhomogeneities need to be considered for each object and require individual calibration measurements to be corrected. Minor deviations of the gradient timing is of little effect if k-space is encoded in parallel lines of identical direction (spin warp imaging). The corresponding global shift leads only to a change in the image phase with no effect in magnitude reconstruction. If the k-space acquisition direction is varying, however, such as in EPI or radial acquisitions, even minor trajectory shifts due to timing errors or eddy currents lead to data inconsistencies that need to be considered in the reconstruction.
A frequently underestimated effect is the flip angle variation over the slice profile in 2D slice selective imaging. The signal can deviate significantly from the one calculated for the nominal flip angle leading to variations in contrast.