More than One RF-Pulse: Echoes & Phase Graphs
Matthias Weigel1,2

1Division of Radiological Physics, Dept. of Radiology, University Hospital Basel, Basel, Switzerland, 2Dept. of Biomedical Engineering, University of Basel, Basel, Switzerland


The basic ideas and the resulting potential of the Extended Phase Graph (EPG) concept are described. It represents an elegant means for the pictorial and quantitative depiction of the resulting magnetization response in multi pulse sequences. EPGs also aid in the understanding and classification of echo generation. Based on these powerful properties and possibilities, the EPG concept has got a lot of attention during the last years. Additionally, the syllabus provides a collection of known and less known references.


Nowadays, modern MRI sequences like multi spin echo or steady state approaches apply a multitude of gradients and RF pulses. The extended phase graph (EPG) concept represents an elegant means for the pictorial and quantitative depiction of the resulting magnetization response in such multi pulse sequences (1–17). It aids in the understanding and classification of echo generation and allows quantitating echo intensities efficiently. Relaxation effects are trivial to be included (3–5,7–11,13,14). It is also possible to consider more complex or advanced phenomena like flow or diffusion (7,10,13,18).

Based on these powerful possibilities, the EPG concept has got a lot of attention during the last years. This educational talk presents the basic ideas and the resulting potential. The syllabus mentions key and advanced aspects and provides a collection of known and less known references.

Extended Phase Graphs – Key Aspects of the Concept

The EPG concept - also known as the configuration model, simply phase graphs, or under other “reinventions” - is mainly based on two key frameworks: First, the original phase graph approach as suggested by Woessner and Kaiser (1,2) and, second, the reinterpretation by means of Fourier space as described by Hennig (3–5). Particularly, it is the combination of the two that makes the EPG concept so powerful. The educational review (13) emphasizes this fact, for instance.

Extended Phase Graphs – The Fourier Representation

The Bloch equation is well suited for (graphical) magnetization response simulations or for considering complex physical MR effects; however, it is less practical for the quantitation of echo intensities or for the prediction of echo times.

One aspect that makes the Bloch equation approach inefficient is that, typically, the evolution of thousands of isochromats has to be determined over time. Then, the vector sum of all isochromats is calculated. This is neither computationally efficient nor an exact solution.

A switch into Fourier space of both the longitudinal and transverse gives a serious improvement: the complete magnetization constellation of the ensemble can be depicted by a set of dedicated Fourier components, which are also called configurations. Thus, all actions of the RF pulses, gradients and other physical observables on the magnetization are just represented by the action on these configurations. Hence, this approach is accurate, simple, efficient and good to code into software.

Extended Phase Graphs – The Phase Graph Method

The phase graph technique was originally published as the partition state method and works in position space (1,2). It has been developed to calculate signal amplitudes in a spin echo experiment. Most importantly, phase graph algorithms are based on the insight that the action of an RF pulse with flip angle α on transverse magnetization Mtrans can be described as a superposition of three magnetization parts:

  1. A part Mtrans−deph, which is proportional to cos2(α/2), remaining unaffected (0deg pulse).
  2. A part Mtrans−ref, which is proportional to sin2(α/2), being refocused to form a spin echo later on (180deg pulse).
  3. A part Mlong, which is converted into spatially modulated z-magnetization. It can be retrieved by a later RF pulse to form a stimulated echo.

Such a phase graph is very practical to predict the time points of echo formation after a series of gradients and RF pulses.

In an EPG, the same observation is true for the Fourier configuration states introduced above. As a result, a complete MRI sequence is simply depicted as a series of evolving configuration states that split up with each RF pulse (3–5,7–11,13,16,17).

Advanced Descriptions like Non-periodic EPGs or Handling Diffusion

Since a constructive interference of magnetization is desired, MRI sequences are usually periodic with identical dephasing patterns. This effect leads to simple (but practical) and periodic EPGs with fully dephased configuration states of multiples of 2π. Echoes are depicted by not dephased states called F0. Typical sequence representatives are many variants of steady state sequences (SSFP) and turbo spin echoes (TSE). For this, dedicated example software can be downloaded from Weigel’s EPG website (19), see also Ref. (13). Furthermore, a common stumbling block in software coding and a closer look at the EPG Fourier domains are elaborated in abstract (15).

The implementation of diffusion or other motion phenomena necessitates considering the exact dephasing history, i.e., the exact gradient patterns and sequence over time (10,13). Simplifications like a rectangular gradient can cause non-negligible changes in diffusion sensitivity as demonstrated for a Hyperecho diffusion preparation (20) in abstract (21). Since particularly diffusion effects are also present in magnetization preparations, which often have non-periodic and irregular gradient dephasings and RF pulse spacings, overall, an EPG and software framework that can handle these effects is frequently desired.

Ref. (10) already showed that it is indeed possible to define generalized EPGs with varying slopes that can account for the above described effects. Programming this into software is somewhat more demanding. A possible realization for such non-periodic and exact EPGs that also include advanced effects like diffusion is the EPGspace framework provided by Weigel (13,19,22), also hosted at It proved success in the calculation of diffusion sensitivities and effective b-factors for TSE sequences with realistic gradient patterns on all three spatial encoding axes (23) or for variants of Hyperecho diffusion preparations (21).

The EPG generalizations require also a solution for partially dephased configuration states such that the macroscopic net magnetization can be calculated at all time. Based on his older ideas, Weigel explained in a recent article how this requirement can be solved (19,24). In short, a field-of-view (FOV) is presumed and the EPG configurations states generally contribute each a sinc-modulated net magnetization in dependence of their total dephasing angle φ. The contributions vanish quickly for higher dephased configuration states. This approach was also included in EPGspace (19,22).

As a side note and to demonstrate a limitation, a framework like EPGspace can indeed calculate accumulating diffusion weightings along the phase encoding direction; however, any EPG approach can basically never assess realistic echo intensities with included phase encoding: This would necessitate knowing the internal object structure! The EPG concept presumes a homogeneous, infinite object, i.e., a delta-peak in k-space (13).


No acknowledgement found.


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Proc. Intl. Soc. Mag. Reson. Med. 26 (2018)