Matthias Weigel^{1,2}

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*.

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.

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 **M _{trans}** can be described as a
superposition of three magnetization parts:

- A part
**M**, which is proportional to cos_{trans−deph}^{2}(α/2), remaining unaffected (0deg pulse). - A part
**M**, which is proportional to sin_{trans−ref}^{2}(α/2), being refocused to form a spin echo later on (180deg pulse). - A part
**M**, which is converted into spatially modulated z-magnetization. It can be retrieved by a later RF pulse to form a stimulated echo._{long}

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).

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 F_{0}. 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 *Bitbucket.org*. 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).

1. Woessner DE. Effects of Diffusion in Nuclear Magnetic Resonance Spin-Echo Experiments. J Chem Phys 1961;34:2057–2061.

2. Kaiser R, Bartholdi E, Ernst RR. Diffusion and Field-Gradient Effects in NMR Fourier Spectroscopy. J Chem Phys 1974;60:2966–2979.

3. Hennig J. Multiecho imaging sequences with low refocusing flip angles. J Magn Reson 1988;78:397–407.

4. Hennig J. Echoes - how to generate, recognize, use or avoid them in MR-imaging sequences. Part I. Conc Magn Reson 1991;3:125–143.

5. Hennig J. Echoes - how to generate, recognize, use or avoid them in MR-imaging sequences. Part II. Conc Magn Reson 1991;3:179–192.

6. Sobol WT, Gauntt DM. On the stationary states in gradient echo imaging. J Magn Reson Imaging 1996;6:384–98.

7. Sodickson A, Cory DG. A generalized k-space formalism for treating the spatial aspects of a variety of nmr experiments. Prog Nucl Mag Res Sp 1998;33:77–108.

8. Scheffler K. A Pictorial Description of Steady-States in Rapid Magnetic Resonance Imaging. Conc Magn Reson 1999;11:291–304.

9. Vlaardingerbroek MT, den Boer JA. Magnetic Resonance Imaging. Berlin: Springer Verlag; 2004.

10. Weigel M, Schwenk S, Kiselev VG, Scheffler K, Hennig J. Extended phase graphs with anisotropic diffusion. J Magn Reson 2010;205:276–85.

11. Hargreaves, Brian A., Miller, Karla L. Using Extended Phase Graphs: Review and Examples. In: Proceedings of the 21st Annual Meeting of ISMRM 2013:3718.

12. Malik SJ, Padormo F, Price AN, Hajnal JV. Spatially resolved extended phase graphs: modeling and design of multipulse sequences with parallel transmission. Magn Reson Med 2012;68:1481–94.

13. Weigel M. Extended phase graphs: Dephasing, RF pulses, and echoes - pure and simple. J Magn Reson Imaging 2015;41:266–295.

14. Ganter C. Steady state of gradient echo sequences with radiofrequency phase cycling: analytical solution, contrast enhancement with partial spoiling. Magn Reson Med 2006;55:98–107.

15. Weigel M. Extended Phase Graphs. In: Proceedings of the 23th Annual Meeting of ISMRM 2015:5640.

16. Weigel M. Extended Phase Graphs and the Analysis of Echo Formation. In: Proceedings of the 18th Annual Meeting of ISMRM 2010:Weekend Educational MR Physics for Physicists.

17. Weigel M. Spin Phase Graphs & the Analysis of Echo Formation & Signal Contrast. In: Proceedings of the 19th Annual Meeting of ISMRM 2011:Weekend Educational MR Physics for Physicists.

18. Kiselev VG. Calculation of diffusion effect for arbitrary pulse sequences. J Magn Reson 2003;164:205–211.

19. Weigel M. Available Software for Extended Phase Graph (EPG) Simulation. http://epg.matthias-weigel.net.

20. Hennig J, Schulte AC, Il’Yasov K, Speck O, Kiselev V. Hyperecho Diffusion Imaging: A New Look at Diffusion Mechanisms. Proc Intl Soc Magn Reson Med 2002;10:433.

21. Schwenk S, Weigel M, Kiselev VG, Hennig J. The Influence of Trapezoidal Gradient Shape on the b-factor of Hyperecho Diffusion Weighted Sequences. In: Proceedings of the 18th Annual Meeting of ISMRM 2010:1605.

22. Weigel M. EPGspace: A Flexible Framework for Extended Phase Graphs of Periodic and Non-Periodic MRI Sequences. In: Proceedings of the 24th Annual Meeting of ISMRM 2016:3195.

23. Weigel M, Hennig J. Diffusion sensitivity of turbo spin echo sequences. Magn Reson Med 2012;67:1528–1537.

24. Weigel M. Assessing the Macroscopic Net Magnetization in Arbitrary and Generalized Extended Phase Graphs. arXiv 2018.